Research Article Lunar CubeSat Impact Trajectory...

Research ArticleLunar CubeSat Impact Trajectory Characteristics asa Function of Its Release Conditions

Young-Joo Song1 Ho Jin2 and Ian Garick-Bethell23

1Satellite Ground System Development Team Satellite Operation Division Korea Aerospace Research Institute169-84 Gwahagno Yuseong-Gu Daejeon 305-806 Republic of Korea2School of Space Research Kyung Hee University 1 Seocheon-dong Giheung-gu Yongin-si Gyeonggi-do 446-701 Republic of Korea3Department of Earth and Planetary Sciences University of California Santa Cruz 1156 High Street Santa Cruz CA 95064 USA

Correspondence should be addressed to Ho Jin benhokhuackr

Received 26 October 2014 Revised 25 February 2015 Accepted 1 March 2015

Academic Editor Gongnan Xie

Copyright copy 2015 Young-Joo Song et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a part of early system design activities trajectory characteristics for a lunar CubeSat impactor mission as a function of itsrelease conditions are analyzedThe goal of this mission is to take measurements of surface magnetic fields to study lunar magneticanomalies To deploy the CubeSat impactor a mother-ship is assumed to have a circular polar orbit with inclination of 90 degrees ata 100 km altitude at the Moon Both the in- and out-of-plane direction deploy angles as well as delta-V magnitudes are consideredfor the CubeSat release conditions All necessary parameters required at the early design phase are analyzed including CubeSatflight time to reach the lunar surface impact velocity cross ranges distance and associated impact angles which are all directlyaffected by the CubeSat release conditions Also relative motions between these two satellites are analyzed for communication andnavigation purposes Although the current analysis is only focused on a lunar impactor mission the methods described in thiswork can easily be modified and applied to any future planetary impactor missions with CubeSat-based payloads

1 Introduction

Since the first launch of a CubeSat in 2003 more thanone hundred CubeSats have been put in orbit [1] IndeedCubeSats have been proven to enable extremely low-costmissions in near Earth orbit with greater launch accessibilityRecently ideas to apply CubeSat technology to deep spaceexploration concepts have greatly increased [2] Over thecoming decade it is expected that diverse science returnscould be obtained from extremely low-cost solar systemexploration missions with improved CubeSats technologiesthat are beyond those demonstrated to date [3] Recentlythe NASA Innovative Advanced Concepts (NIAC) programselected interplanetary CubeSats for further investigation toenable a new class ofmissions beyond lowEarth orbit [4]Thepotential missions initially considered by NIAC are MineralMapping of an Asteroid Solar System Escape TechnologyDemonstration Earth-Sun Sub-L1 Space Weather MonitorPhobos Sample Return Earth-Moon L2 Radio Quiet Obser-vatory and Out-of-Ecliptic Missions [3] Other than these

missions innovative deep space exploration concepts usingCubeSats have also been proposed or studied For examplea CubeSat on an Earth-Mars free-return trajectory couldcharacterize the hazardous radiation environment beforehuman mission to Mars and watch for potential hazardousNear Earth Objects (NEOs) [5] Another mission proposesto study asteroid regolith mechanics and primary accretionprocesses [6]

Since 1992 Korea has been continuously operating morethan ten Earth-orbiting satellites and is now expanding itsinterests to planetary missions The Korean space programhas plans to launch a lunar orbiter and lander around 2020and also has plans to explore Mars asteroids and deep spacein the future Therefore the Korean aeronautical and spacescience community has performed numerous relatedmissionstudies and the Korea Aerospace Research Institute (KARI)is performing pre-phase work for the lunar mission Severalpreliminary design studies have already been conducted suchas an optimal transfer trajectory analysis [7ndash16] mappingorbit analysis [17] landing trajectory analysis [18 19] contact

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 681901 16 pageshttpdxdoiorg1011552015681901

2 Mathematical Problems in Engineering

schedule analysis [20 21] link budget analysis [22ndash24]landerrover system design [25ndash29] and candidate payloaddesign analysis [30] Before 2020 the first Korean lunarpathfinder orbiter mission is scheduled for launch around2017 with international collaboration This may not onlysecure critical technologies but also form a solid basis for thenext lunar mission around 2020 The Korean lunar sciencecommittee is now working to select the main scientificobjectives for the lunar orbiter mission and one of thecandidates is to fly a CubeSat impactor to explore lunarmagnetic anomalies and associated albedo features knownas swirls [31]

In 1959 the Soviet spacecraft called ldquoLuna-1rdquo carried amagnetometer to the Moon Data from ldquoLuna-1rdquo concludedthat theMoon has no global magnetic field like Earthrsquos How-ever from the Apollo 15 and 16 missions it was discoveredthat strongly magnetized materials are distributed all overthe Moonrsquos crust The origin of lunar magnetism is one ofthe oldest problems that is still debated in the field of lunarscience [32] Understanding the origin of swirls may helpto understand not only geological processes but also spaceweathering effects on the lunar surface Although previouslunar missions such as the National Aeronautics and SpaceAdministrationrsquos (NASA) Lunar Prospector and the JapanAerospace Exploration Agencyrsquos (JAXA) KAGUYA have alsomeasured lunar magnetic fields these data are not sufficientto completely characterize magnetic anomaly regions sincethey were obtained at high altitudes (gt20 km) [32] Afterthe completion of its nominal mission a lunar orbiter couldbe crashed into a target area to take measurements at lowaltitudes just as most of the past lunar orbiters have endedtheirmissions However such an impact is not only expensivebut also has rare launch accessibility and mostly there areenormousmission demands that need to be performed at lowaltitude

For this reason a new idea is to use a CubeSat carrying amagnetometer as a payload and impact at the target regionof interest Actually the concept of CubeSat impactor tomeasure lunar magnetic fields near the surface has alreadybeen discussed [32] In [32] two major lunar transfer scenar-ios are proposed to deliver the CubeSat impactor The firstoption is to use the Planetary HitchHiker (PHH) conceptwhich is a small spacecraft designed to be accommodatedas a secondary payload on a variety of launch vehicles Inthis concept the launch vehicle places the PHH spacecraftinto Geostationary Transfer Orbit (GTO) to reduce missioncosts and after insertion into GTO the PHH spacecraft usesonboard propulsion to cruise to theMoon and finally releasethe CubeSat impactor after appropriate orbital conditionsare established Appropriate orbital conditions to deploy theCubeSat impactor will be established by several Lunar OrbitInsertions (LOI) orbit adjustments and station-keepingburns as conventional lunar mission sequences The secondconcept is to board the CubeSat impactor into a geosta-tionary spacecraft as a payload and deploy it after reachinggeostationary orbit (GEO) The released impactor will spiralout to the Moon with its own minimized ion propulsionsystem and upon entering the Moonrsquos gravitational sphereof influence the CubeSat will directly impact the target area

without entering lunar orbit However these two missionscenarios have several challenging aspects to overcome forexample longer flight times to reach the lunar orbit (which isexpected to be more than 100 days) tolerating large amountsof radiation exposure even though the mission starts fromGEO and most importantly establishing a shallow impactangle (lt10 deg) during the impact phase to meet the scienceobjectives which is a more critical factor if a mother-ship isnot used [32]

Another promisingmethod to achieve thismission objec-tive might be to fly the lunar CubeSat impactor as oneof the scientific payloads on the lunar orbiter We believethat this approach will partly ease the challenging aspectsthat have been raised in the previously discussed scenariosespecially establishing the very shallow impact angle Indeedone of the major expected contributions of using CubeSats inplanetary missions is that a large variety of near-surface sci-ence experiments could be performed [2] Most of planetaryexploration to date has been achieved through remote sensingfrom orbiters or surface exploration by landers Howeversuch methods are expensive and risky and the science datagathered from the near-surface can be limited [2] For thesereasons CubeSat payload planetary missions are vigorouslysuggested not only for the Moon but also for Mars Europaand other deep space exploration missions For exampleNASA is considering launching a CubeSat-based payload ona futureMars explorationmission around 2016 to early 2020susing excess capacity on the missionrsquos primary spacecraft[33] Recently the Jet Propulsion Laboratory (JPL) selectedand funded CubeSat concept studies for the NASA missionEuropaClipperThismission is aimed for launch around 2025withmultiple CubeSats and science objectives for the releasedCubeSats in the Jovian system would include reconnaissancefor future landing sites gravity fields magnetic fields atmo-spheric and plume science and radiation measurements toenhance our understanding of Europa [34]

As we have already developed experimental CubeSatswhich are Triplet Ionospheric Observatory CubeSat for IonNeutral Electron MAgnetic fields (TRIO CINEMA) andScientific CubeSat with Instruments for Global Magneticfield and rAdiation (SIGMA) [31] their heritage can directlybe applied to the lunar impactor mission proposed in thisstudy For the TRIO CINEMA mission now in Earth orbit(CINEMA 1was launched in 2012 andCINEMAs 2 and 3werelaunched in 2013) three different institutions collaboratedto develop the CINEMA series UC Berkeley for the 1stCINEMA and Kyung Hee University for the 2nd and 3rdCINEMAsTheCINEMAmission usesmagnetometers basedon Magneto Resistive (MR) sensors developed by ImperialCollege [35]The SIMGACubeSat funded by the 2013KoreanCubeSat program from KARI aims to test a magnetome-ter in low Earth orbit based on a Miniaturized FluxGate(MFG) [31] The appropriate magnetometer for the proposedCubeSat impactor mission would be selected through theseheritage instruments For the data transfer between theCubeSat impactor and a mother-ship we have determinedthat Ultra High Frequency (UHF) band communication isfeasible and a separation on the night side of theMoonwould

Mathematical Problems in Engineering 3

be preferred but not the essential factor to avoid solar windperturbations to the magnetic field [36]

Themain objective of this paper is to perform a feasibilitystudy to obtain numerous insights into a proposed lunarCubeSat impactor mission especially analyzing impact tra-jectory characteristics and their dependence on release con-ditions from amother-ship Furthermore the authors believethat these preliminary impact trajectory design studies willbe helpful for further detailed system definition and designactivities Therefore all necessary parameters for the earlydesign phase are analyzed CubeSat flight time before impactat the lunar surface impact velocity cross ranges distancewhich is measured on the lunar surface and associatedimpact angles which are all directly affected by the CubeSatimpactor release conditions Based on our results benefitsand drawbacks are discussed for selected impact trajectoriesRelative motions between a mother-ship and the CubeSatimpactor are also analyzed for communication and naviga-tion system design purposes and through the analysis manychallenging aspects are identified that have to be resolved ata further detailed mission design stage Although the currentanalysis only considers a lunar CubeSat impactormission themethods can easily be modified and applied to other similarmissions where CubeSats are released from a mother-shiporbiting around another planet and will certainly have broadimplications for future planetary missions with CubeSats InSection 2 system dynamics such as equations of motion theclosest approach condition derivation impact angle calcula-tion and relative motion geometry between a mother-shipand the impactor are described to simulate a given impactormission Section 3 provides detailed numerical implicationsand presumptions that are made for the current study andvarious simulation results with further studies planned arepresented through Section 4 In Section 5 conclusions aresummarized with discussions of work that is planned to beperformed in the near future

2 System Dynamics

21 Equations of Motion Two-body equations of motion ofthe CubeSat impactor after release from a mother-ship flyingin the vicinity of the Moon can be expressed as

[

RCube

VCube] =

[[

[

VCube

minus

120583RCube

R3Cube

]]

]

(1)

with initial conditions of

[

RCube (0)

VCube (0)] = [

RSC (119905119903)

VSC (119905119903) + ΔV

] (2)

where 120583 is the gravitational constant of the Moon 119905119903is the

time of the CubeSat impactor release and R and V denoteposition and velocity vectors expressed in theMoon-centeredMoon Mean Equator and IAU vector of epoch J2000 (M-MME2000) frame In each vector subscripts ldquoSCrdquo and ldquoCuberdquoindicate the mother-ship and the CubeSat impactor respec-tively ΔV is the divert delta-119881 expressed in M-MME2000

frame which is generated during the impactor deploymentprocessΔV can be expressed asΔV = ΔVPOD+ΔVTST whereΔVPOD is the delta-119881 induced from the Poly PicosatelliteOrbital Deployer (P-POD) and ΔVTST is the delta-119881 inducedfrom a thruster mounted on the CubeSat Indeed ΔVPODand ΔVTST should be regarded separately however only theoverall impulsive ΔV is considered in the current simulationfor a preliminary analysisThe overallΔV can be transformedfrom a defined delta-119881 vector expressed in the mother-shiprsquosLocal VerticalLocal Horizontal (LVLH) frame ΔVLVLH asfollows

ΔV = [Q1] ΔVLVLH

(3)

whereQ1is the direction cosine matrix defined as [37]

Q1

=

[[[[[

[

((minusRSC times VSC) times (minusRSC)1003816100381610038161003816(minusRSC times VSC) times (minusRSC)

1003816100381610038161003816

)

119909SC

(minusRSC times VSC1003816100381610038161003816minusRSC times VSC

1003816100381610038161003816

)

119909SC

(minusRSC1003816100381610038161003816minusRSC1003816100381610038161003816

)

119909SC


1003816100381610038161003816

)

119910SC


1003816100381610038161003816

)

119910SC


)

119910SC


1003816100381610038161003816

)

119911SC


1003816100381610038161003816

)

119911SC


)

119911SC

]]]]]

]

(4)

In (4) subscripts 119909SC 119910SC and 119911SC denote the unit vectorcomponent of defined RSC in M-MME2000 frame In addi-tion ΔVLVLH can also be expressed with the unit vectors asfollows [38]

ΔVLVLH= Δ119881 cos (120572 (119905

119903)) cos (120573 (119905

119903)) iLVLHSC

+ Δ119881 sin (120572 (119905119903)) cos (120573 (119905

119903)) jLVLHSC

+ Δ119881 sin (120573 (119905119903))

kLVLHSC

(5)

where Δ119881 is the divert delta-119881 magnitude and iLVLHSC jLVLHSC and kLVLHSC are the defined comoving unit vectors in the trans-verse opposite normal and along inward radial directions allattached to the mother-ship The unit vectors are defined asfollows kLVLHSC is the unit vector which always points from themother-shiprsquos center of themass along the radius vector to theMoonrsquos center jLVLHSC is the unit vector that lies in the oppositedirection of the mother-shiprsquos angular momentum vectorand iLVLHSC is the unit transverse vector perpendicular to bothkLVLHSC and jLVLHSC that points in the direction of the mother-shiprsquos velocity vector In Figure 1 the defined geometry of theCubeSat impactor release conditions from the mother-ship isshown

At the time of the CubeSat impactor release 119905119903 the

defined in-plane direction delta-119881 deploy angle 120572(119905119903) is

measured from the unit vector iLVLHSC to the projected vectoronto the local horizontal plane that is perpendicular tothe orbital plane The out-of-plane delta-119881 direction deployangle 120573(119905

119903) is measured from the local horizontal plane to

the delta-119881 vector in the vertical direction Therefore angles120572(119905119903) and 120573(119905

119903) can be regarded as the mother-shiprsquos ldquoyawrdquo

and ldquopitchrdquo attitude orientation angle at the release moment


To Moonrsquos center

CubeSat

Mother-shipMother-ship

flying direction

CubeSatflying direction

119829SC

Δ119829

LVLHSC

LVLHSC

LVLHSC

120573(tr)

120572(tr)

119825SC

Figure 1 Defined geometry of the CubeSat impactor release conditions from the mother-ship (not to scale)

22 The Closest Approach Condition After separation theclosest approach condition between the CubeSat and thelunar surface is computed using the method described in[39] which finds the root of a numerically integrated singlenonlinear equation For numerical integration the Runge-Kutta-Fehlberg 7-8th order variable step size integrator isused During the root-finding process the objective function119891obj is given as follows

119891obj (119905app) =1003816100381610038161003816ℎCube

1003816100381610038161003816 (6)

Utilizing methods described in [39] the CubeSatimpactorrsquos closest approach time to the lunar surface 119905appand the associated areodetic altitude ℎCube can be computedby determining whether the given objective function isincreasing or decreasing with user specified lower 119905lowapp andupper 119905upapp bounds of time search interval and convergencecriterion 120576root In this study ℎCube is computed assuming alunar flatting coefficient after conversion of the CubeSatrsquosstates expressed in the M-MME2000 frame into the MoonMean Equator and Prime Meridian (M-MMEPM) frameIf ℎCube is found to be greater than 0 km then the CubeSatwill not impact the lunar surface and will fly over withdetermined ℎCube at 119905app If ℎCube is determined to be 0 kmthen there will be a lunar surface impact at 119905app and thus119905app can be regarded as the CubeSat impact time 119905imp or theCubeSat flight time (CFT) In addition at 119905imp the areodeticlongitude and latitude of the impact point (120582(119905imp) 120601(119905imp))

120579(ti)

h(ti)

h(tr)

(120582(timp) 120601(timp)) d(ti) (120582(ti) 120601(ti)) (120582(tr) 120601(tr))

Lunar surface

CubeSat

Impactpoint

Mother-shiporbit

Impact trajectory

Mother-ship

Figure 2 Geometry of the defined CubeSat impact angle (not toscale)

can be easily obtained from the states components expressedin the M-MMEPM frame

23 The CubeSat Impact Angle The CubeSat impact angle120579(119905119894) can be approximated as 120579(119905

119894) = tanminus1(ℎ(119905

119894)119889(119905119894)) [32]

where ℎ(119905119894) is the mother-shiprsquos areodetic altitude at every

instant of moment 119905119894 during the impact phase which has

ranges of 119905119903

le 119905119894

lt 119905imp Also 119889(119905119894) is the cross range

distance between the subground point where the areodeticaltitude is measured at (120582(119905

119894) 120601(119905119894)) and the impact point

(120582(119905imp) 120601(119905imp)) Thus the cross range distance can be


regarded as ldquotravel distancerdquo of the CubeSat measured onthe lunar surface after separation In Figure 2 the definedCubeSat impact angle geometry is shown [32] The 119889(119905

119894)

between the (120582(119905119894) 120601(119905119894)) and (120582(119905imp) 120601(119905imp)) is computed

using the method described in [40] as follows

119889 (119905119894)

= 119903pol ([(1 + 119891 + 1198912) 120575]

+ 119903Moon [[(119891 + 1198912) sin 120575 minus (

1198912

2

) 1205752 csc 120575]]

+ 120585 [minus(

119891 + 1198912

2

) 120575 minus (

119891 + 1198912

2

)

sdot sin 120575 cos 120575 + (

1198912

2

) 1205752 cot 120575]

+ 1199032

Moon [minus(

1198912

2

) sin 120575 cos 120575]

+ 1205852[(

1198912

16

) 120575 + (

1198912

16

) sin 120575 cos 120575

minus(

1198912

2

) 1205752 cot 120575 minus (

1198912

8

) sin 120575cos3120575]

+ 119903Moon120585 [(

1198912

2

) 1205752 csc 120575 + (

1198912

2

) sin 120575cos2120575])

(7)

In (7) 119903pol and 119903Moon are the Moonrsquos mean polar radius andmean equatorial radius respectively Also 119891 is the Moonrsquosflattening coefficient 119891 = 1 minus (119903pol119903Moon) and 120575 and 120585 arethe defined parameters as follows

120575 = tanminus1(((sin 1205761sin 1205762) + (cos 120576

1cos 1205762) cos 120582)

sdot ((sin 120582 cos 1205762)2

+ [sin (1205932minus 1205931)

+2 cos 1205762sin 1205761sin2 (120582

2

)]

2

)

minus12

)

120585 = 1 minus (

(cos 1205761cos 1205762) sin 120582

sin 120575

)

2

(8)

where

120582 = 120582 (119905imp) minus 120582 (119905119894)

1205761= tanminus1 (

119903pol sin120601 (119905119894)

119903Moon cos120601 (119905119894)

)

1205762= tanminus1(

119903pol sin120601 (119905imp)

119903Moon cos120601 (119905imp))

1205932minus 1205931= (120601 (119905imp) minus 120601 (119905

119894))

+ 2 [sin (120601 (119905imp) minus 120601 (119905119894))]

sdot [(120578 + 1205782+ 1205783) 119903Moon minus (120578 minus 120578

2+ 1205783) 119903pol]

120578 = (

119903Moon minus 119903pol

119903Moon + 119903pol)

(9)

24 Relative Motion between the Mother-Ship and CubeSatAfter the CubeSat impactor is deployed relative motionbetween the mother-ship and CubeSat impactor during theimpact phase in M-MME2000 frame can be determined asin

RSC2C = RSC minus RCube

VSC2C = VSC minus VCube

RC2SC = RCube minus RSC

VC2SC = VCube minus VSC

(10)

In (10) RSC2C and VSC2C are the CubeSat impactorrsquosposition and velocity vectors seen from a mother-ship andRC2SC and VC2SC are the mother-shiprsquos position and velocityvectors seen from theCubeSat impactor during impact phaserespectively Using (11)sim(14) RSC2C VSC2C and RC2SC VC2SCcan be expressed with respect to the LVLH frame attached tothemother-shipRLVLH

SC2C VLVLHSC2C and the LVLH frame attached

to the CubeSat impactor RLVLHC2SC V

LVLHC2SC respectively

RLVLHSC2C = [Q

1]119879RSC2C (11)

VLVLHSC2C = [Q

1]119879VSC2C (12)

RLVLHC2SC = [Q

2]119879RC2SC (13)

VLVLHC2SC = [Q

2]119879VC2SC (14)

In (13) and (14) Q2is the direction cosine matrix

that transforms a vector from LVLH frame attached to theCubeSat impactor to the CubeSat centered MME2000 framedefined in (15) In (15) subscripts 119909Cube 119910Cube and 119911Cubedenote the unit vector components of RCube defined in theM-MME2000 frame


Q2=

[[[[[[[[[[

[

(

(minusRCube times VCube) times (minusRCube)1003816100381610038161003816(minusRCube times VCube) times (minusRCube)

1003816100381610038161003816

)

119909Cube

(

minusRCube times VCube1003816100381610038161003816minusRCube times VCube

1003816100381610038161003816

)

119909Cube

(

minusRCube1003816100381610038161003816minusRCube

1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

]]]]]]]]]]

]

(15)

Using (11)ndash(14) the relative location between the mother-ship and CubeSat impactor during the impact phase 119905 =

119905imp minus 119905119903 can simply be expressed in terms of in-plane

120572(119905)SC2C and out-of-plane direction angle 120573(119905)SC2C of theCubeSat impactor seen from the mother-ship Also in-plane120572(119905)C2SC and out-of-plane angles 120573(119905)C2SC of the mother-ship are seen from the CubeSat impactor as shown in (16) InFigure 3 geometry of defined relative motions between themother-ship and CubeSat impactor is shown

120572 (119905)SC2C = tanminus1((RLVLH

SC2C

10038161003816100381610038161003816RLVLHSC2C

10038161003816100381610038161003816)119895LVLHSC

(RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816)119894LVLHSC

)

120573 (119905)SC2C = sinminus1((

RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816

)

119896LVLHSC

)

120572 (119905)C2SC = tanminus1((RLVLH

C2SC

10038161003816100381610038161003816RLVLHC2SC

10038161003816100381610038161003816)119895LVLHCube

(RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816)119894LVLHCube

)

120573 (119905)C2SC = sinminus1((

RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816

)

119896LVLHCube

)

(16)

3 Numerical Implication and Presumptions

During simulations several assumptions aremade to simplifythe given problem as to focus on early design phase analysisTwo-body equations of motion are used to propagate boththe mother-ship and the CubeSat impactor and the divertdelta-119881 to separate the CubeSat impactor is assumed to be animpulsive burn as already discussed In addition themother-shiprsquos attitude is assumed to immediately reorient to itsnominal attitude just after the CubeSat impactor deploymentActually reorientation of the mother-shiprsquos attitude may takedifferent durations dependent on its attitude control strategywhich is certainly another parameter that must be consideredat the system design level For numerical integration theRunge-Kutta-Fehlberg 7-8th order variable step size integra-tor is usedwith truncation error tolerance of 120576 = 1times10

minus12TheJPLrsquos DE405 is used to derive the accurate planetsrsquo ephemeris[41] that is the Earth and the Sun positions seen from theMoon with all planetary constants To convert coordinatesystems between the M-MME2000 and M-MMEPM framesthe lunar orientation specified by JPL DE405 is used forhigh precision work to be performed in the near future

The current simulation assumes the CubeSat impactor isdeployed at the moment when the mother-ship is flyingover the north polar region of the Moon with a circular90 deg inclined polar orbit of 100 km altitude Thereforeinitial orbital elements of the mother-ship expressed in theM-MME2000 frame are given as semimajor axis of about18382 km zero eccentricity 90 deg inclination 0 deg of rightascension of ascending node and finally 90 deg of argumentof latitude As the operational altitude is expected to be100 km for Korearsquos first experimental lunar orbiter missionthe current analysis only considers the 100 km altitude caseHowever additional analysis regarding various mother-shipaltitudes could be easily made by simple modifications ofthe current method The initial epoch of CubeSat impactorrelease is assumed to be July 1 2017 corresponding to Korearsquosfirst experimental lunar orbiter mission Most importantlythe CubeSat impactor release conditions are assumed to beas follows For release directions the out-of-plane releasedirection is increased from 90 deg to 180 deg in steps of05 deg which indicate that the CubeSat impactor will alwaysbe deployed in the opposite direction of the mother-shipflight direction in order to not interfere the mother-shiprsquosoriginal flight path For in-plane release directions they areconstrained to always have 0 deg to avoid plane changes dur-ing the impact phase Even though the CubeSat is assumed toalways be deployed in the opposite direction of the mother-ship flight direction with 0 deg in-plane release directionsthe released CubeSat can be impacted to any location on thelunar surface as the ground track of the mother-ship willmap the entire lunar surface with its assumed 90 deg orbitalinclination For divert delta-119881magnitudes they are increasedfrom 0ms to 90ms with 05ms steps For the closestapproach conditions derivation the convergence criterion isgiven as 120576root = 1 times 10

minus12 and lower 119905lowapp and upper 119905upappbounds of the time search interval are given to be 0min and118min respectively By applying these constraints the Cube-Sat impactor will impact the lunar surface within one orbitalperiod of the mother-shiprsquos orbit (about 118min for 100 kmaltitude at the Moon) which will ease the communicationlink problem between the mother-ship and the CubeSatimpactor Also during the cross range distance computationthe Moonrsquos oblate effect is regarded with flatting coefficientof about 119891 = 00012 During the following discussionsseveral figures are expressed in normalized units for the easeof interpretation and all parameters used to normalize unitsare explained in detail at corresponding subsections


Mother-ship

Mother-shiporbit

Impact trajectory

Impact point

Moon center

CubeSat

119829SC

119829Cube

LVLHSC

LVLHSC

LVLHSC

120573(t)SC2C

120572(t)SC2C

119825SC

120573(t)C2SC

120572(t)C2SC

LVLHCube

LVLHCube

LVLHCube

119825Cube

119825LVLHC2SC 119825LVLH

SC2C

Figure 3 Relative geometry between the mother-ship and the CubeSat impactor during the impact phase (not to scale)

4 Simulation Results

41 Impact Trajectory Characteristics Analysis

411 Impact Opportunities as a Function of Release Condi-tions In this subsection the CubeSat impact opportunitiesas a function of release conditions are analyzed The impactopportunities are directly analyzed by using computed closestapproach altitudes between the CubeSat impactor and thelunar surface Corresponding results are depicted in Figure 4FromFigure 4 it can be easily noticed that there are specific ofout-of-plane deploy angles and divert delta-119881 ranges that leadthe CubeSat to impact the lunar surface When the out-of-plane deploy angle is about 180 deg the CubeSat can impactwith the minimum amount of divert delta-119881 magnitudeabout 235ms This result indicates that the CubeSat shouldbe released in exactly the opposite direction of the mother-shiprsquos velocity direction tominimize the required divert delta-119881 magnitude which is a quite general behavior Howeverthe minimum magnitude of about 235ms is quite a largeamount to be supported only by the P-POD separationmech-anism For example if performance equivalent to P-PODMkIII is under consideration about 215ms of additional divert

90100

110120

130140

150160

170180

010203040

50607080

90

0

20

40

60

80

100

Clos

est a

ppro

ach

altit

ude (

km)

Impact region

Nonimpact region

Out-of-plane deploy angle (deg)Divert delta-V (ms)

Figure 4 CubeSat impact opportunities when released from amother-ship having 100 km altitude and a 90 deg inclined circularorbit around the Moon


Table 1 List of required divert delta-Vs with respect to arbitraryselected out-of-plane deploy angles

Out-of-planedeploy angle (deg)

Min divertdelta-V (ms)

Max divertdelta-V (ms)

90 More than 90 NA100 6750 9000110 5150 9000120 4100 9000130 3400 9000140 2950 9000150 2700 9000160 2500 9000170 2400 9000180 2350 9000

delta-119881 should be supported since the CubeSatrsquos exit velocityfrom the P-POD Mk III is approximately 20ms for a 4 kgCubeSat [42] In addition ejection velocity limitations due tothe mother-ship system configuration should additionally beconsidered

Another fact discovered is that if the out-of-plane deployangle is more than 140 deg the rate of degradation in totalrequired divert delta-119881magnitude tends to be decreased thatis only about 6ms difference on divert delta-119881 magnitudewhile about 40 deg of out-of-plane deploy angles are changed(140 deg deploy with about 295ms and 180 deg deploy withabout 235ms) With a maximum magnitude of assumeddivert delta-119881 90ms the CubeSat can only impact the lunarsurface with about 915 deg of out-of-plane deploy angleThus if the out-of-plane deploy angle is less than 915 degmore than 90ms of divert delta-119881 is required In Table 1specific ranges of required divert delta-119881 magnitude areshown with arbitrary selected out-of-plane deploy anglesOther than these release conditions shown in Table 1 theCubeSat will not impact the lunar surface and thus willrequire additional analysis with different assumptions onrelease conditions For example if the CubeSat is releasedwith about 180 deg out-of-plane deploy angle with about2ms of divert delta-119881 the CubeSat will not be on courseto hit the lunar surface rather it will attain an ellipticalorbit having about 9311 km of perilune altitude (the closestapproach distance) which is only about 7 km of reducedaltitude compared to the initial mother-shiprsquos altitude

412 Impact Parameter Characteristics The CubeSat impactparameters during the impact phase are analyzed throughthis subsection including time left to impact after deployingfrom the mother-ship cross range distance impact angleand velocity at the time of impact Among these param-eters the time left to impact after deployment from themother-ship can be regarded as CFT In Figure 5 char-acteristics of derived parameters are shown with releaseconditions that guaranteed the CubeSat impact With givenassumptions on release conditions CFTs are found to be

within ranges of 1566min (deployed with 130 deg of out-of-plane angle with 90ms divert delta-119881) to 5600min(deployed with 180 deg of out-of-plane angle with 235msdivert delta-119881) For cross range distances they tend toremain within the range of 146607 km (deployed with13550 deg and 9000ms) to 540880 km (deployed with180 deg and 2350ms) and for impact angles they tend toremain within the range of 108 deg (deployed with 180 degand 2350ms) to 398 deg (deployed with 13550 deg and9000ms) For impact velocities it is found that they remainwithin 164ms (deployed with 180 deg and 2350ms) and172 kms (deployed with 9150 deg and 9000ms) respec-tively To compute the impact angle discussed above crossrange distance is computed with the subsatellite point wherethe CubeSat is released (120582(119905

119903) 120601(119905119903)) Thus the resultant

impact angle could be slightly changed as it is dependenton the point where the cross ranges are measured duringevery moment of the impact phase The resultant impactangles 108sim398 deg sufficiently satisfy the impact anglerequirement for the proposed impactor mission which isgiven as less than 10 deg [32] Note that the impact anglecondition is constrained for this mission instead of theduration of time spent over the target area to measure thelunar magnetic field as the spatial coverage at low altitudeis more important than the measurement duration This isbecause the onboard magnetometer can measure at a rapidpace to fulfill the science goal For example one of thestrong candidate impact sites Reiner Gamma [32 43 44]has a spatial extent of about 70 times 30 km and when themagnetic field is measured at 200Hz with an impact velocityof sim2 kms [32] then the spatial resolution is just 10mwhich is more than sufficient By analyzing CFTs (shownin Figure 5(a)) the CubeSatrsquos power subsystem requirementespecially battery capacity requirement during the impactphase could be determined as the impact is planned tooccur on the night side of the Moon to obtain scientificallymeaningful data by avoiding solar wind interference [36]Recall that this simulation is performed under conditionsthat the CubeSat should impact the lunar surface within oneorbital period of the mother-shiprsquos orbit (about 118min for100 km altitude at the Moon) indicating that the releasedCubeSat cannot orbit the Moon This assumption is madeto ease the communication architecture design between themother-ship and the CubeSat impactor Therefore it canbe easily noticed that the discovered CFTs are all less than118min and also the derived cross range distances are lessthan about 10915 km which is the circumference of theMoon Although we only considered a mother-ship with analtitude of 100 km our results revealed several challengingaspects that should be solved in further detailed designstudies

42 Examples of Impact Cases

421 Impact Trajectory Analysis The characteristics of threemajor representative example impact trajectories Cases AB and C are analyzed in this subsection Case A representsthe case when the CubeSat is released from a mother-ship with 1300 deg of out-of-plane angle with 9000ms


015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)

Out-of-plane angle (deg) Divert delta-V (ms)

(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)

Out-of-plane angle (deg) Divert delta

-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)

Out-of-plane angle (deg) Divert delta-V

(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)

Figure 5 CubeSat impact parameter characteristics during the impact phase (a) CubeSat flight time (b) cross range distance (c) impactangle and (d) impact velocity

divert delta-119881 Case B is the case released with 16450 degwith 3150ms of divert delta-119881 and finally Case C isthe case released with 18000 deg of out-of-plane angle with2350ms divert delta-119881 respectively Note that Cases A andC are the minimum (about 1566min) and maximum (about5592min) CFT cases out of the entire simulation cases andCase B is selected as an example which has 3583min ofCFT the average CFT between Cases A and C Actuallyamong all solutions there were three different cases having3583min of CFTwith different release conditions 16450 degwith 3150ms 15050 deg with 3200ms and 13900 degwith 3400ms Among these three cases the case withminimum divert delta-119881 with 3150ms is selected for CaseB In Figure 6 associated impact trajectories are shown withnormalized distance units Lunar Unit (LU) where 1 LU isabout 17382 km In addition Cases A B and C shown in

Figure 6 can be understood as short- medium- and long-arccases respectively

For Case A the CubeSat is released at June 1 2017000000 (UTC) and impact occurred at June 1 2017 001540(UTC) At the time of the CubeSat release the mother-shiprsquosvelocity is found to be about 163 kms and the CubeSatimpacted with about 167 kms For Cases B and C boththe CubeSat release time and mother-shiprsquos velocity at thetime of release are the same as with Case A but the impacttime is found to be June 1 2017 003550 (UTC) with animpact velocity of about 169 kms for Case B For Case C theimpact time is found to be about June 1 2017 005558 (UTC)with an impact velocity of about 170 kms As expectedalthough less divert delta-119881 was applied (9000ms for CaseA 3150ms for Case B and 2350ms for Case C) a fasterimpact velocity is achieved as CFT becomes longer that is


minus1

minus05

0

05

1

minus1 minus05 0 05 1

minus10

1

X (LU)Y (LU)

Z(L

U)

Impacttrajectory Release point

Mother-shipMother-ship orbit

at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)


Mother-shipMother-shiporbitat impact

(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)

Figure 6 Selected example of CubeSat impact trajectories (a) is for Case A (b) is for Case B and (c) is for Case C

Case C is about 30ms faster than Case A as the CubeSatis more accelerated In addition for the same reasonswe discovered greater separations on relative ground trackpositions between themother-ship and the CubeSat impactornear the time of impact By analyzing the CFT the impactlocation can roughly be estimated For example for Case Cthe impact location will be near the south pole of the Moonas the derived CFT is almost half of the mother-shiprsquos orbitalperiod (about 118min) as shown in Figure 6(c) When onlyregarding the divert delta-119881rsquos magnitude which still requiresadditional support from an onboard thruster to contributethe remainder of the delta-119881 it seems that Case C wouldbe the appropriate choice for the proposed impact missionHowever if the CubeSat power availability is considered asthe major design driver Case B would be the proper choice

which has a CFT of about 3583min This is due to the factthat the current design for the CubeSat impactor missionconsiders operation only with a charged battery (currentpower subsystem is expected to support maximum of about30min) during the impact phase However Case B stillrequires more divert delta-119881 which is another critical aspectthat has to be satisfied by means of sustained system designstudies

In Figure 7 ground tracks for the selected three CubeSatimpact trajectories are shown In Figure 7 as already dis-cussed it can be clearly seen that the relative ground locationsbetween the mother-ship and the CubeSat impactor broadenat the time of impact with longer CFTs The positions of theEarth and Sun at the time of impact are also depicted inFigure 7 which could aid in further detailed mission studies


Releasepoint

Impact pointfor Case A

Mother-ship at impactfor Case C

Impact pointfor Case C

Mother-ship at impactfor Case B

Impact point for Case B

Mother-ship at impactfor Case A Mother-ship

ground track

Impact ground trackfor Cases A B and C

Earth location at impactfor Cases A B and C

Sun location at impactfor Cases A B and C

minus200 minus150 minus100 minus50 0 50 100 150 200minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)

Figure 7 Ground tracks for the selected CubeSat impact trajecto-ries Cases A B and C with positions of the Sun and Earth at theimpact time

that is the Earth communication opportunities for themother-ship during the impact phase as well as determiningwhether the impact would occur on the day or night side It isexpected that further detailed analysis could be easily madebased on the current analysis

422 Impact Angle Analysis In this subsection impact anglevariations during the impact phase for three different examplecases (Cases A B and C) are analyzed Before discussingimpact angle variations the CubeSat altitude variationsduring the impact phase are firstly analyzed as shown inFigure 8(a) In Figure 8(a) the 119909-axis denotes normalizedremaining cross range distance before the impact and the 119910-axis represents areodetic altitude for each case In Figure 8(a)cross range distance is normalized with 147135 km for CaseA 341177 km for Case B and 540429 km for Case Crespectively To normalize areodetic altitudes for every case10209 km is used As expected areodetic altitude almostlinearly decreases as cross range approaches zero for CaseA However for Cases B and C areodetic altitudes do notlinearly decrease but decrease with small fluctuations whencompared to Case A The main cause of this phenomenon isdue to the shape of the resultant CubeSat impact trajectoryFor example if impact had not occurred near perilunefor Case C then the resultant impact trajectory would beconsidered an elliptical orbit around the Moon and thusthe depicted behaviors of altitude variations for given impacttrajectories (Cases B and C) are rather general results

Based on altitude variations (shown in Figure 8(a)) theimpact angle variations are analyzed as shown in Figure 8(b)In Figure 8(b) the 119909-axis indicates the remaining time toimpact the lunar surface expressed in normalized time unitsand the 119910-axis indicates the CubeSat impact angle expressed

in deg To normalize time units 1566min is used for Case A3583min for Case B and 5597min for Case C respectivelyAt the beginning of the impact phase at the CubeSat releasetime the impact angle for Case A is found to be about397 deg for Case B about 172 deg and for Case C about108 deg respectively The impact angle increases linearlyto about 466 deg for Case A however not surprisinglydifferent trends are observed for Cases B and C For CaseB the impact angle increases to about 209 deg 1947minafter release and decreases linearly for the remainder of theimpact phase (ending at about 175 deg at impact) For CaseC the impact angle increases to about 121 deg 1378minafter the CubeSat release and decreases to about 005 degat the final impact time Areodetic altitudes achieved by theCubeSat before about 10 km cross range distance apart fromthe impact point are as follows about 81991m for Case Aabout 30833m for Case B and only about 8m for CaseC are derived respectively If the achieved impact angle isconsidered as the major design driver Case A would be abetter option than Case B or C as it achieves higher impactangle during the impact phase since there is uncertainty inlunar surface heights Indeed the resultant relations betweendivert delta-119881 magnitude and impact angle shown throughthis subsection would be another major issue to be dealtwith in further detailed trade-off design studies In Table 2detailedmission parameters obtained for three different casesare summarized Note that every eastern longitude and everynorthern latitude shown in Table 2 is based on M-MMEPMframe and associated altitude and cross range distances areall in an areodetic reference frame

423 Relative Motion Analysis During the impact phaserelative motion between the mother-ship and the CubeSatimpactor is one of themajor factors that has to be analyzed forthe communication architecture design In Figure 9 relativemotion characteristics are shown for three different impactcases 119909-axes in subfigures of Figure 9 are expressed innormalized time units and 119910-axes indicate in-plane (left sideof Figure 9) andout-of-plane (right side of Figure 9) directionangles in deg To normalize time units 1566min is used forCase A (shown at Figures 9(a) and 9(b)) 3583min for CaseB (shown at Figures 9(c) and 9(d)) and 5592min for CaseC (shown at Figures 9(e) and 9(f)) respectively In additionsolid lines represent relative locations of themother-ship seenfrom the CubeSat (ldquoC to Mrdquo in the following discussions)and dotted lines represent the location of the CubeSat seenfrom themother-ship (ldquoM toCrdquo in the following discussions)By investigating in-plane direction motions (left side of theFigure 9) variations it is observed that a ldquophase shiftrdquo betweenthemother-ship and the CubeSat occurred during the impactphase in every simulated case As an example of Case A in-plane angle remained to be about 180 deg (for M to C) or0 deg (for C to M) at the early phase of the CubeSat releaseand then switched to be 0 deg (for M to C) and 180 deg(for C to M) for the remaining time of impact phase Thisresult indicates that the mother-ship will fly ahead of theCubeSat if seen from the CubeSat itself or will fly behindthe mother-ship if seen from the mother-ship during about1193min (herein after the phase I) Then the ldquophase shiftrdquo


Table 2 Detailed mission parameters derived for Cases A B and C

Parameters Short-arc case (Case A) Medium-arc case (Case B) Long-arc case (Case C)Release time 2017-06-01 000000 (UTC) larr larr

Impact time 2017-06-01 001540 (UTC) 2017-06-01 003550 (UTC) 2017-06-01 005558 (UTC)Mother-ship velocity at release 163 (kms) larr larr

CubeSat velocity at impact 167 (kms) 169 (kms) 170 (kms)CubeSat time of flight 1566 (min) 3583 (min) 5592 (min)Release location

Longitude 17285 (deg) larr larr

Latitude 8939 (deg) larr larr

Altitude 10209 (km) larr larr

Impact locationLongitude minus14992 (deg) minus14962 (deg) minus13314 (deg)Latitude 4103 (deg) minus2306 (deg) minus8870 (deg)Altitude 000 (km) larr larr

Mother-ship at impactLongitude minus14993 (deg) minus14964 (deg) minus14744 (deg)Latitude 4173 (deg) minus1999 (deg) minus8148 (deg)Altitude 10092 (km) 10024 (km) 10204 (km)

Earth at impactLongitude 744 (deg) 744 (deg) 744 (deg)Latitude minus076 (deg) minus080 (deg) minus083 (deg)

Sun at impactLongitude 10362 (deg) 10344 (deg) 10327 (deg)Latitude minus150 (deg) minus150 (deg) minus150 (deg)

Cross distance range (release to impact) 147135 (km) 341177 (km) 540429 (km)Impact angle

At release 397 (deg) 172 (deg) 108 (deg)At about 10 km cross range distance before impact 466 (deg) 186 (deg) 004 (deg)

0Normalized cross range distance

Case A

Case B

Case C

minus1 minus09 minus08 minus07 minus06 minus05 minus04 minus03 minus02 minus010

02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)

0Normalized time before impact


Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)

Figure 8 Altitude (a) and impact angle (b) variations and during the impact phase for Cases A B and C

occurs which indicates the relative flight position is changedduring the remainder of time (herein after the phase II)about 373minThus about 7618 of the entire impact phasewill be phase I for Case A For Case B phase I is foundto be about 2016min and 1067min for phase II such thatthe phase I portion is about 6539 Finally for Case Cphase I is found to be about 2389min and 3208min forphase II with the phase I portion equal to 4268 Theseresults indicate that as the CFT gets longer the portion

of phase I will be less (or phase II will be more) due tothe dynamic behavior of the CubeSat in the impact phaseas previously discussed in Section 421 For out-of-planedirection relative motions similar behaviors were obtainedThe existence of the phase shift could also be confirmedthrough the sign of rate changes at the peaks of out-of-planedirection variations about minus8999 deg for C to M and about8999 deg for M to C as shown in Figures 9(b) 9(d) and9(f) Note that the out-of-plane direction angle for C to M


0

50

100

150

200

Normalized time since CubeSat release

In-p

lane

dire

ctio

n (d

eg)

Case A

0minus1 minus08 minus06 minus04 minus02

(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50

Normalized time since CubeSat release0minus1 minus08 minus06 minus04 minus02

Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)

Figure 9 Relativemotion between themother-ship andCubeSat during impact phase In-plane (left) and out-of-plane (right) direction anglevariations for Case A ((a) and (b)) Case B ((c) and (d)) and Case C ((e) and (f))


00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)

0Normalized time since CubeSat release

Case ACase BCase C


(b)

Figure 10 Relative range (a) and velocity (b) variation histories between the mother-ship and the CubeSat during the three different (CasesA B and C) impact cases

is always negative as the mother-ship will always be locatedabove the defined local horizontal plane of theCubeSat frameAs expected the initial value of the out-of-plane directionangle variations is directly related to the out-of-planeCubeSatrelease direction However as analyses in this study are madeunder the assumption of instantaneous attitude reorientationof the mother-ship after CubeSat separation the resultsshown in this subsectionmay vary based on themother-shiprsquosattitude control strategy especially for out-of-plane relativedirections For additional analysis relative range and velocityvariations between the mother-ship and CubeSat during theimpact phase are investigated for three different examplecases as shown in Figure 10 Note that variation historiesshown in Figure 10 are all C to M cases Without a doubtrelative distances between the mother-ship and CubeSatincrease as time reaches the impact time and Case C showedthe largest separation distance at the time of impact about24882 km than the other two cases about 10329 km for CaseA and 13866 km for Case B For relative velocity variationsas expected the final relative velocity of Case C showedthe maximum about 22491ms with the largest differencebetween the initial release and final relative velocity at impactFor Case A the final relative velocity between the mother-ship and the CubeSat is found to be about 15491ms and15781ms for Case B Results provided in this subsection areexpected to be used as a basis for the detailed communicationsystem design that is optimum onboard antenna location tomaximize the communication performance between the twosatellites during the impact phase

43 Further Analysis Planned As the current study is per-formed as a part of early system design activities a futurestudy will be carried out for more detailed mission analysisTo provide enough divert delta-119881 magnitude alternativeapproaches will be taken into account such as use of aminiaturized CubeSat thruster during the impact phase tocompensate the insufficient delta-119881 from P-POD separationat a lower altitude as well as with enhanced P-POD delta-119881performance that we can possibly achieve while satisfying the

requirement on shallow impact angle Therefore numeroustrade-off studies will be made in further analyses Duringthe numerous trade-off studies the effect of the perturbingforces due to the nonsphericity of the Moon and 3rd bodies(eg the Earth) to the CubeSat impact trajectory will also beanalyzed in detail The optimization of impact trajectory andattitude control strategy of the CubeSat impactor will also beconsidered Additionally analysis with a specified target (ieReinerGamma) impact area and additional diagnostics of themother-shiprsquos orbital elements at the time of CubeSat releasewill be performedMost importantly tolerable deploy delta-119881errors (not only themagnitude but also the directions) to landin an ellipse of a given target area will be analyzed Throughthis analysis the minimum requirements on the CubeSatonboard propulsion system and numerous insights into themother-shiprsquos orbit and attitude determination accuracy canbe obtained By regarding allowable delta-119881 errors of theCubeSat onboard propulsion system and P-PODmechanismwe could also place some constraints on the CubeSat releasetime For example only small delta-119881 errors will be acceptedif theCubeSat is released very early before impact in contrastrelatively large delta-119881 errors are permitted if the CubeSatis released very close to the impact time to achieve therequired impact accuracy In addition to correct delta-119881errors separation of the deorbit burn strategy into 2sim3 stagesormore could be regarded during the detailedmission designphase Although several challenging aspects remain andstill require sustained research preliminary analysis resultsobtained from this study will give numerous insights into thedesign field of planetary impactor missions with CubeSat-based payloads

5 Conclusions

As a part of preliminary mission design and analysis activi-ties the trajectory characteristics of a lunar CubeSat impactorreleased from a lunar orbiter a mother-ship are analyzedin this study The mother-ship is assumed to have a circularpolar orbit with an inclination of 90 degrees at a 100 km


altitude at the Moon Two release conditions are appliedto separate the CubeSat the eject direction (in-plane andout-of-plane with respect to mother-shiprsquos LVLH frame) andthe divert delta-119881rsquos magnitude as these are major factors thatdetermine the flight path of the CubeSat impactor As a resultthe CubeSat impact opportunities are analyzed with relatedmission parameters appropriate release directions divertdelta-119881s magnitude CubeSat flight times impact velocitiescross ranges and impact angles In addition the relative flightmotion between the mother-ship and the CubeSat duringthe impact phase is analyzed to support detailed commu-nication system design activities It is found that the lunarimpactor and its trajectory characteristics strongly dependon the divert delta-119881 magnitude rather than the appliedrelease directions Within the release conditions that we haveassumed the CubeSat flight time after separation takes about1566sim5600min with about 164sim172 kms impact velocityAlso the cross range (travel distance) on a lunar groundtrack was found to be about 146607sim540880 km with animpact angle of about 108sim398 deg It is confirmed that thevery shallow impact angle (less than 10 deg) can be achievedwith the proposed CubeSat impactor release scenarios whichis a critical requirement to meet the science objectivesHowever the requiredminimumdivert delta-119881magnitude toimpact the CubeSat is found to be 235ms and this is quitelarge compared to the capabilities of the current availableP-POD system From relative motion analysis it is foundthat there is a phase-shift stage between the mother-shipand the CubeSat during the impact phase and the momentof this phase-shift is strongly dependent on the CubeSatflight timeThis indicates that the onboard antenna locationsshould be optimized to maximize the communication per-formance within limited power sources during the impactphase Although this analysis is made using basic dynamicsand several assumptions additional guidelines for furthermission design and analysis are well defined from the currentresults and a future study will be carried out formore detailedmission analysis Also it is expected that the analysismethodsdescribed in this work can easily be modified and appliedto any other similar future planetary impactor missions withCubeSat-based payloads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the BK21 Plus and NRF-2014M1A3A3A02034761 Program through the NationalResearch Foundation (NRF) funded by the Ministry ofEducation and the Ministry of Science ICT and FuturePlanning of South Korea

References

[1] M Swartwout ldquoThe first one hundred CubeSats a statisticallookrdquo Journal of Small Satellites vol 2 no 2 pp 213ndash233 2013

[2] A T Klesh and J C Castillo-Rogez ldquoApplications of NanoSatsto planetary explorationrdquo in Proceedings of the AIAA SPACEConference amp Exposition Pasadena Calif USA September2012

[3] R L Staehle D Blaney H Hemmati et al ldquoInterplanetaryCubeSats opening the solar system to a broad community atlower costrdquo Journal of Small Satellites vol 2 no 1 pp 161ndash1862013

[4] D L Blaney R L Staehle B Betts et al ldquoInterplanetaryCubeSats small low cost missions beyond low Earth Orbitrdquo inProceedings of the 43rd Lunar and Planetary Science Conferencep 1868 Woodlands Tex USA March 2012

[5] J Vannitsen B Segret J J Maiu and J-C Juang ldquoCubeSat onEarth-Mars Free Return Trajectory to study radiation hazardsin the future manned missionrdquo in Proceedings of the EuropeanPlanetary Science Congress (EPSC rsquo13) EPSC2013-1088 LondonUK September 2013

[6] EAsphaug and JThangavelautham ldquoAsteroid regolithmechan-ics and primary accretion experiments in a CubeSatrdquo in Pro-ceedings of the 45th Lunar and Planetary Science Conference p2306 Woodlands Tex USA March 2014

[7] Y J Song S Y Park K H Choi and E S Sim ldquoDevelopmentof Korean preliminary lunar mission design softwarerdquo Journalof The Korean Society for Aeronautical amp Space Sciences vol 36no 4 pp 357ndash367 2008

[8] Y J Song S Y Park K H Choi and E S Sim ldquoOptimal earth-moon trajectory design using constant and variable low thrustrdquoJournal of the Korean Society for Aeronautical Space Sciences vol37 no 9 pp 843ndash854 2009

[9] Y-J Song S-Y Park K-H Choi and E-S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[10] Y J Song JWoo S Y Park KHChoi andE S Sim ldquoThe earthmoon transfer trajectory design and analysis using intermediateloop orbitsrdquo Journal of Astronomy and Space Sciences vol 26 no2 pp 171ndash186 2009

[11] J Woo Y-J Song S-Y Park H-D Kim and E-S Sim ldquoAnearth-moon transfer trajectory design and analysis consideringspacecraftrsquos visibility from daejeon ground station at TLI andLOI maneuversrdquo Journal of Astronomy and Space Science vol27 no 3 pp 195ndash204 2010

[12] Y J Song S Y Park H D Kim J H Lee and E S Sim ldquoTransLunar Injection (TLI)maneuver design and analysis using finitethrustrdquo Journal of The Korean Society for Aeronautical amp SpaceSciences vol 38 no 10 pp 998ndash1011 2010

[13] T SNo andG E Jeon ldquoA study on optimal earth-moon transferorbit design using mixed impulsive and continuous thrustrdquoJournal of the Korean Society for Aeronautical amp Space Sciencesvol 38 no 7 pp 684ndash692 2010

[14] D H Lee and H C Bang ldquoLow thrust fuel optimal earthescape trajectories designrdquo Journal of the Korean Society forAeronautical amp Space Sciences vol 35 no 7 pp 647ndash654 2007

[15] Y-J Song S-Y Park H-D Kim J-H Lee and E-S SimldquoAnalysis of delta-V losses during lunar capture sequence usingfinite thrustrdquo Journal of Astronomy and Space Science vol 28no 3 pp 203ndash216 2011

[16] S Choi Y Song J Bae E Kim and G Ju ldquoDesign and analysisof korean lunar orbiter mission using direct transfer trajectoryrdquoThe Journal of the Korean Society for Aeronautical amp SpaceSciences vol 41 no 12 pp 950ndash958 2013

[17] Y-J Song S-Y Park H-D Kim and E-S Sim ldquoDevelopmentof precise lunar orbit propagator and lunar polar orbiterrsquos


lifetime analysisrdquo Journal of Astronomy and Space Science vol27 no 2 pp 97ndash106 2010

[18] D H Cho B Y Jeong D H Lee and H C Bang ldquoOptimalperilune altitude of lunar landing trajectoryrdquo InternationalJournal of Aeronautical and Space Sciences vol 10 no 1 pp 67ndash74 2009

[19] B Y Jeong Y H Choi S J Jo and H C Bang ldquoTerrain aidedinertial navigation for precise planetary landingrdquo Journal of theKorean Society for Aeronautical amp Space Sciences vol 38 no 7pp 673ndash683 2010

[20] Y-J Song S-I Ahn S-J Choi and E-S Sim ldquoGround contactanalysis for korearsquos fictitious lunar orbiter missionrdquo Journal ofAstronomy and Space Science vol 30 no 4 pp 255ndash267 2013

[21] Y J Song S J Choi S I Ahn and E S Sim ldquoAnalysis ontracking schedule and measurements characteristics for thespacecraft on the phase of lunar transfer and capturerdquo Journalof Astronomy and Space Sciences vol 31 no 1 pp 51ndash61 2014

[22] S Kim D Yoon and K Hyun ldquoGround stations of korean deepspace network for lunar explorationsrdquoThe Journal of the KoreanSociety for Aeronautical amp Space Sciences vol 38 no 5 pp 499ndash506 2010

[23] W Lee K Cho D Yoon and K Hyun ldquoDesign and perfor-mance analysis of downlink in space communications systemfor lunar explorationrdquo Journal of Astronomy and Space Sciencevol 27 no 1 pp 11ndash20 2010

[24] W Lee D Yoon and J Lee ldquoPerformance analysis of maximumdata rate for telemetry links in space communications for lunarexplorationsrdquo Journal of the Korean Society for Aeronautical ampSpace Sciences vol 39 no 1 pp 42ndash49 2011

[25] Y-K Kim H-D Kim J-H Lee E-S Sim and S-W JeonldquoConceptual design of roverrsquos mobility system for ground-basedmodelrdquo Journal of Astronomy and Space Sciences vol 26 no 4pp 677ndash692 2009

[26] W-S Eom Y-K Kim J-H Lee G-H Choi and E-S SimldquoStudy on a suspension of a planetary exploration rover toimprove driving performance during overcoming obstaclesrdquoJournal of Astronomy and Space Science vol 29 no 4 pp 381ndash387 2012

[27] T J Son K S Na J H Lim K W Kim and D S HwangldquoDevelopment of a structure for lunar lander demonstratorrdquoAerospace Engineering andTechnology vol 12 no 1 pp 213ndash2202013

[28] W B Lee and D Y Rew ldquoVirtual flight test for conceptual lunarlander demonstratorrdquo Aerospace Engineering and Technologyvol 12 no 1 pp 87ndash93 2013

[29] S S Yang Y C Kang J Y Son M H Oh J H Kim andJ Y Cho ldquoOptimization of shock absorption system for lunarlander considering the effect of lunar regolithrdquo The Journal ofthe Korean Society for Aeronautical amp Space Sciences vol 42 no4 pp 284ndash290 2014

[30] K J Kim J-H Lee H Seo et al ldquoAn introduction to the lunarand planetary science activities in Koreardquo Advances in SpaceResearch vol 54 no 10 pp 2000ndash2006 2014

[31] H J Lee J K Lee S-M Baek et al ldquoA CubeSat mission forKorean lunar explorationrdquo in Proceedings of the 45th Lunar andPlanetary Science Conference p 1783 Woodlands Tex USAMarch 2014

[32] I Garrick-Bethell R P Lin H Sanchezd et al ldquoLunarmagneticfield measurements with a cubesatrdquo in Sensors and Systemsfor Space Applications VI vol 8739 of Proceedings of SPIEBaltimore Md USA April 2013

[33] T Komarek Z Bailey H Schone T Jedrey and A ChandlerldquoNovel ideas for exploring Mars with CubeSats challenges andpossibilitiesrdquo in Proceedings of the 10th IAA Low-Cost PlanetaryMissions Conference Pasadena Calif USA June 2013

[34] NASA Jet Propulsion Laboratory ldquoJPL Selects Europa CubeSatProposals for Studyrdquo October 2014 httpwwwjplnasagovnewsnewsphpfeature=4330

[35] H Jin J Seon K H Kim et al ldquoThe system design of TRIOcinema missionrdquo in Proceedings of the 38th COSPAR ScientificAssembly p 12 Bremen Germany July 2010

[36] H Jin I Garrick-Bethell Y J Song et al ldquoPreliminary analysisof a lunar CubeSat impactor missionrdquo in Proceedings of theLunar Science Workshop p 13 Yongin Republic of Korea May2014

[37] D A Vallado Fundamentals of Astrodynamics and ApplicationsKluwer Academic Publishers 3rd edition 2007

[38] H D Curtis Orbital Mechanics for Engineering Students Else-vier Aerospace Engineering Series Butterworth-Heinemann2nd edition 2009

[39] R P Brent Algorithms for Minimization without DerivativesDover 2002

[40] EM Sodano ldquoGeneral non-iterative solution of the inverse anddirect geodetic problemsrdquo Bulletin Geodesique vol 39 no 1 pp69ndash89 1965

[41] E M Standish ldquoJPL planetary and lunar ephemerides DE405LE405rdquo Jet Propulsion Laboratory Interoffice MemorandumIOM 312F-98-048 1998

[42] W Lan R Munakata L R Nugent and D Pignatelli ldquoPolypicosatellite orbital deployer Mk III Rev E user guiderdquo TheCubeSat Program CP-PPODUG-10-1 California PolytechnicState University 2007

[43] L L Hood and G Schubert ldquoLunar magnetic anomalies andsurface optical propertiesrdquo Science vol 208 no 4439 pp 49ndash51 1980

[44] DHemingway and I Garrick-Bethell ldquoMagnetic field directionand lunar swirl morphology insights from Airy and ReinerGammardquo Journal of Geophysical Research Planets vol 117 no10 Article ID E10012 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


schedule analysis [20 21] link budget analysis [22ndash24]landerrover system design [25ndash29] and candidate payloaddesign analysis [30] Before 2020 the first Korean lunarpathfinder orbiter mission is scheduled for launch around2017 with international collaboration This may not onlysecure critical technologies but also form a solid basis for thenext lunar mission around 2020 The Korean lunar sciencecommittee is now working to select the main scientificobjectives for the lunar orbiter mission and one of thecandidates is to fly a CubeSat impactor to explore lunarmagnetic anomalies and associated albedo features knownas swirls [31]

In 1959 the Soviet spacecraft called ldquoLuna-1rdquo carried amagnetometer to the Moon Data from ldquoLuna-1rdquo concludedthat theMoon has no global magnetic field like Earthrsquos How-ever from the Apollo 15 and 16 missions it was discoveredthat strongly magnetized materials are distributed all overthe Moonrsquos crust The origin of lunar magnetism is one ofthe oldest problems that is still debated in the field of lunarscience [32] Understanding the origin of swirls may helpto understand not only geological processes but also spaceweathering effects on the lunar surface Although previouslunar missions such as the National Aeronautics and SpaceAdministrationrsquos (NASA) Lunar Prospector and the JapanAerospace Exploration Agencyrsquos (JAXA) KAGUYA have alsomeasured lunar magnetic fields these data are not sufficientto completely characterize magnetic anomaly regions sincethey were obtained at high altitudes (gt20 km) [32] Afterthe completion of its nominal mission a lunar orbiter couldbe crashed into a target area to take measurements at lowaltitudes just as most of the past lunar orbiters have endedtheirmissions However such an impact is not only expensivebut also has rare launch accessibility and mostly there areenormousmission demands that need to be performed at lowaltitude

For this reason a new idea is to use a CubeSat carrying amagnetometer as a payload and impact at the target regionof interest Actually the concept of CubeSat impactor tomeasure lunar magnetic fields near the surface has alreadybeen discussed [32] In [32] two major lunar transfer scenar-ios are proposed to deliver the CubeSat impactor The firstoption is to use the Planetary HitchHiker (PHH) conceptwhich is a small spacecraft designed to be accommodatedas a secondary payload on a variety of launch vehicles Inthis concept the launch vehicle places the PHH spacecraftinto Geostationary Transfer Orbit (GTO) to reduce missioncosts and after insertion into GTO the PHH spacecraft usesonboard propulsion to cruise to theMoon and finally releasethe CubeSat impactor after appropriate orbital conditionsare established Appropriate orbital conditions to deploy theCubeSat impactor will be established by several Lunar OrbitInsertions (LOI) orbit adjustments and station-keepingburns as conventional lunar mission sequences The secondconcept is to board the CubeSat impactor into a geosta-tionary spacecraft as a payload and deploy it after reachinggeostationary orbit (GEO) The released impactor will spiralout to the Moon with its own minimized ion propulsionsystem and upon entering the Moonrsquos gravitational sphereof influence the CubeSat will directly impact the target area

without entering lunar orbit However these two missionscenarios have several challenging aspects to overcome forexample longer flight times to reach the lunar orbit (which isexpected to be more than 100 days) tolerating large amountsof radiation exposure even though the mission starts fromGEO and most importantly establishing a shallow impactangle (lt10 deg) during the impact phase to meet the scienceobjectives which is a more critical factor if a mother-ship isnot used [32]

Another promisingmethod to achieve thismission objec-tive might be to fly the lunar CubeSat impactor as oneof the scientific payloads on the lunar orbiter We believethat this approach will partly ease the challenging aspectsthat have been raised in the previously discussed scenariosespecially establishing the very shallow impact angle Indeedone of the major expected contributions of using CubeSats inplanetary missions is that a large variety of near-surface sci-ence experiments could be performed [2] Most of planetaryexploration to date has been achieved through remote sensingfrom orbiters or surface exploration by landers Howeversuch methods are expensive and risky and the science datagathered from the near-surface can be limited [2] For thesereasons CubeSat payload planetary missions are vigorouslysuggested not only for the Moon but also for Mars Europaand other deep space exploration missions For exampleNASA is considering launching a CubeSat-based payload ona futureMars explorationmission around 2016 to early 2020susing excess capacity on the missionrsquos primary spacecraft[33] Recently the Jet Propulsion Laboratory (JPL) selectedand funded CubeSat concept studies for the NASA missionEuropaClipperThismission is aimed for launch around 2025withmultiple CubeSats and science objectives for the releasedCubeSats in the Jovian system would include reconnaissancefor future landing sites gravity fields magnetic fields atmo-spheric and plume science and radiation measurements toenhance our understanding of Europa [34]

As we have already developed experimental CubeSatswhich are Triplet Ionospheric Observatory CubeSat for IonNeutral Electron MAgnetic fields (TRIO CINEMA) andScientific CubeSat with Instruments for Global Magneticfield and rAdiation (SIGMA) [31] their heritage can directlybe applied to the lunar impactor mission proposed in thisstudy For the TRIO CINEMA mission now in Earth orbit(CINEMA 1was launched in 2012 andCINEMAs 2 and 3werelaunched in 2013) three different institutions collaboratedto develop the CINEMA series UC Berkeley for the 1stCINEMA and Kyung Hee University for the 2nd and 3rdCINEMAsTheCINEMAmission usesmagnetometers basedon Magneto Resistive (MR) sensors developed by ImperialCollege [35]The SIMGACubeSat funded by the 2013KoreanCubeSat program from KARI aims to test a magnetome-ter in low Earth orbit based on a Miniaturized FluxGate(MFG) [31] The appropriate magnetometer for the proposedCubeSat impactor mission would be selected through theseheritage instruments For the data transfer between theCubeSat impactor and a mother-ship we have determinedthat Ultra High Frequency (UHF) band communication isfeasible and a separation on the night side of theMoonwould




2 System Dynamics


[

RCube

VCube] =

[[

[

VCube

minus

120583RCube

R3Cube

]]

]

(1)


[

RCube (0)

VCube (0)] = [

RSC (119905119903)

VSC (119905119903) + ΔV

] (2)




ΔV = [Q1] ΔVLVLH

(3)


Q1

=

[[[[[

[


1003816100381610038161003816

)

119909SC


1003816100381610038161003816

)

119909SC


)

119909SC


1003816100381610038161003816

)

119910SC


1003816100381610038161003816

)

119910SC


)

119910SC


1003816100381610038161003816

)

119911SC


1003816100381610038161003816

)

119911SC


)

119911SC

]]]]]

]

(4)


ΔVLVLH= Δ119881 cos (120572 (119905

119903)) cos (120573 (119905

119903)) iLVLHSC

+ Δ119881 sin (120572 (119905119903)) cos (120573 (119905

119903)) jLVLHSC

+ Δ119881 sin (120573 (119905119903))

kLVLHSC

(5)











CubeSat


flying direction


119829SC

Δ119829

LVLHSC

LVLHSC

LVLHSC

120573(tr)

120572(tr)

119825SC



119891obj (119905app) =1003816100381610038161003816ℎCube

1003816100381610038161003816 (6)


120579(ti)

h(ti)

h(tr)


Lunar surface

CubeSat

Impactpoint

Mother-shiporbit

Impact trajectory

Mother-ship




119894) = tanminus1(ℎ(119905

119894)119889(119905119894)) [32]



ranges of 119905119903

le 119905119894







119894)



119889 (119905119894)

= 119903pol ([(1 + 119891 + 1198912) 120575]

+ 119903Moon [[(119891 + 1198912) sin 120575 minus (

1198912

2

) 1205752 csc 120575]]

+ 120585 [minus(

119891 + 1198912

2

) 120575 minus (

119891 + 1198912

2

)

sdot sin 120575 cos 120575 + (

1198912

2

) 1205752 cot 120575]

+ 1199032

Moon [minus(

1198912

2

) sin 120575 cos 120575]

+ 1205852[(

1198912

16

) 120575 + (

1198912

16

) sin 120575 cos 120575

minus(

1198912

2

) 1205752 cot 120575 minus (

1198912

8

) sin 120575cos3120575]

+ 119903Moon120585 [(

1198912

2

) 1205752 csc 120575 + (

1198912

2

) sin 120575cos2120575])

(7)



1cos 1205762) cos 120582)

sdot ((sin 120582 cos 1205762)2

+ [sin (1205932minus 1205931)

+2 cos 1205762sin 1205761sin2 (120582

2

)]

2

)

minus12

)

120585 = 1 minus (

(cos 1205761cos 1205762) sin 120582

sin 120575

)

2

(8)

where

120582 = 120582 (119905imp) minus 120582 (119905119894)


119903pol sin120601 (119905119894)

119903Moon cos120601 (119905119894)

)

1205762= tanminus1(

119903pol sin120601 (119905imp)

119903Moon cos120601 (119905imp))

1205932minus 1205931= (120601 (119905imp) minus 120601 (119905

119894))

+ 2 [sin (120601 (119905imp) minus 120601 (119905119894))]


2+ 1205783) 119903pol]

120578 = (


119903Moon + 119903pol)

(9)






(10)





RLVLHSC2C = [Q

1]119879RSC2C (11)

VLVLHSC2C = [Q

1]119879VSC2C (12)

RLVLHC2SC = [Q

2]119879RC2SC (13)

VLVLHC2SC = [Q

2]119879VC2SC (14)




Q2=

[[[[[[[[[[

[

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

]]]]]]]]]]

]

(15)





SC2C

10038161003816100381610038161003816RLVLHSC2C

10038161003816100381610038161003816)119895LVLHSC

(RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816)119894LVLHSC

)

120573 (119905)SC2C = sinminus1((

RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816

)

119896LVLHSC

)


C2SC

10038161003816100381610038161003816RLVLHC2SC

10038161003816100381610038161003816)119895LVLHCube

(RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816)119894LVLHCube

)

120573 (119905)C2SC = sinminus1((

RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816

)

119896LVLHCube

)

(16)







Mother-ship

Mother-shiporbit

Impact trajectory

Impact point

Moon center

CubeSat

119829SC

119829Cube

LVLHSC

LVLHSC

LVLHSC

120573(t)SC2C

120572(t)SC2C

119825SC

120573(t)C2SC

120572(t)C2SC

LVLHCube

LVLHCube

LVLHCube

119825Cube


SC2C





90100

110120

130140

150160

170180

010203040

50607080

90

0

20

40

60

80

100

Clos

est a

ppro

ach

altit

ude (

km)

Impact region

Nonimpact region








90 More than 90 NA100 6750 9000110 5150 9000120 4100 9000130 3400 9000140 2950 9000150 2700 9000160 2500 9000170 2400 9000180 2350 9000










015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)


(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)


-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)


(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)






minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2 System Dynamics


[

RCube

VCube] =

[[

[

VCube

minus

120583RCube

R3Cube

]]

]

(1)


[

RCube (0)

VCube (0)] = [

RSC (119905119903)

VSC (119905119903) + ΔV

] (2)




ΔV = [Q1] ΔVLVLH

(3)


Q1

=

[[[[[

[


1003816100381610038161003816

)

119909SC


1003816100381610038161003816

)

119909SC


)

119909SC


1003816100381610038161003816

)

119910SC


1003816100381610038161003816

)

119910SC


)

119910SC


1003816100381610038161003816

)

119911SC


1003816100381610038161003816

)

119911SC


)

119911SC

]]]]]

]

(4)


ΔVLVLH= Δ119881 cos (120572 (119905

119903)) cos (120573 (119905

119903)) iLVLHSC

+ Δ119881 sin (120572 (119905119903)) cos (120573 (119905

119903)) jLVLHSC

+ Δ119881 sin (120573 (119905119903))

kLVLHSC

(5)











CubeSat


flying direction


119829SC

Δ119829

LVLHSC

LVLHSC

LVLHSC

120573(tr)

120572(tr)

119825SC



119891obj (119905app) =1003816100381610038161003816ℎCube

1003816100381610038161003816 (6)


120579(ti)

h(ti)

h(tr)


Lunar surface

CubeSat

Impactpoint

Mother-shiporbit

Impact trajectory

Mother-ship




119894) = tanminus1(ℎ(119905

119894)119889(119905119894)) [32]



ranges of 119905119903

le 119905119894







119894)



119889 (119905119894)

= 119903pol ([(1 + 119891 + 1198912) 120575]

+ 119903Moon [[(119891 + 1198912) sin 120575 minus (

1198912

2

) 1205752 csc 120575]]

+ 120585 [minus(

119891 + 1198912

2

) 120575 minus (

119891 + 1198912

2

)

sdot sin 120575 cos 120575 + (

1198912

2

) 1205752 cot 120575]

+ 1199032

Moon [minus(

1198912

2

) sin 120575 cos 120575]

+ 1205852[(

1198912

16

) 120575 + (

1198912

16

) sin 120575 cos 120575

minus(

1198912

2

) 1205752 cot 120575 minus (

1198912

8

) sin 120575cos3120575]

+ 119903Moon120585 [(

1198912

2

) 1205752 csc 120575 + (

1198912

2

) sin 120575cos2120575])

(7)



1cos 1205762) cos 120582)

sdot ((sin 120582 cos 1205762)2

+ [sin (1205932minus 1205931)

+2 cos 1205762sin 1205761sin2 (120582

2

)]

2

)

minus12

)

120585 = 1 minus (

(cos 1205761cos 1205762) sin 120582

sin 120575

)

2

(8)

where

120582 = 120582 (119905imp) minus 120582 (119905119894)


119903pol sin120601 (119905119894)

119903Moon cos120601 (119905119894)

)

1205762= tanminus1(

119903pol sin120601 (119905imp)

119903Moon cos120601 (119905imp))

1205932minus 1205931= (120601 (119905imp) minus 120601 (119905

119894))

+ 2 [sin (120601 (119905imp) minus 120601 (119905119894))]


2+ 1205783) 119903pol]

120578 = (


119903Moon + 119903pol)

(9)






(10)





RLVLHSC2C = [Q

1]119879RSC2C (11)

VLVLHSC2C = [Q

1]119879VSC2C (12)

RLVLHC2SC = [Q

2]119879RC2SC (13)

VLVLHC2SC = [Q

2]119879VC2SC (14)




Q2=

[[[[[[[[[[

[

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

]]]]]]]]]]

]

(15)





SC2C

10038161003816100381610038161003816RLVLHSC2C

10038161003816100381610038161003816)119895LVLHSC

(RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816)119894LVLHSC

)

120573 (119905)SC2C = sinminus1((

RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816

)

119896LVLHSC

)


C2SC

10038161003816100381610038161003816RLVLHC2SC

10038161003816100381610038161003816)119895LVLHCube

(RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816)119894LVLHCube

)

120573 (119905)C2SC = sinminus1((

RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816

)

119896LVLHCube

)

(16)







Mother-ship

Mother-shiporbit

Impact trajectory

Impact point

Moon center

CubeSat

119829SC

119829Cube

LVLHSC

LVLHSC

LVLHSC

120573(t)SC2C

120572(t)SC2C

119825SC

120573(t)C2SC

120572(t)C2SC

LVLHCube

LVLHCube

LVLHCube

119825Cube


SC2C





90100

110120

130140

150160

170180

010203040

50607080

90

0

20

40

60

80

100

Clos

est a

ppro

ach

altit

ude (

km)

Impact region

Nonimpact region








90 More than 90 NA100 6750 9000110 5150 9000120 4100 9000130 3400 9000140 2950 9000150 2700 9000160 2500 9000170 2400 9000180 2350 9000










015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)


(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)


-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)


(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)






minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











CubeSat


flying direction


119829SC

Δ119829

LVLHSC

LVLHSC

LVLHSC

120573(tr)

120572(tr)

119825SC



119891obj (119905app) =1003816100381610038161003816ℎCube

1003816100381610038161003816 (6)


120579(ti)

h(ti)

h(tr)


Lunar surface

CubeSat

Impactpoint

Mother-shiporbit

Impact trajectory

Mother-ship




119894) = tanminus1(ℎ(119905

119894)119889(119905119894)) [32]



ranges of 119905119903

le 119905119894







119894)



119889 (119905119894)

= 119903pol ([(1 + 119891 + 1198912) 120575]

+ 119903Moon [[(119891 + 1198912) sin 120575 minus (

1198912

2

) 1205752 csc 120575]]

+ 120585 [minus(

119891 + 1198912

2

) 120575 minus (

119891 + 1198912

2

)

sdot sin 120575 cos 120575 + (

1198912

2

) 1205752 cot 120575]

+ 1199032

Moon [minus(

1198912

2

) sin 120575 cos 120575]

+ 1205852[(

1198912

16

) 120575 + (

1198912

16

) sin 120575 cos 120575

minus(

1198912

2

) 1205752 cot 120575 minus (

1198912

8

) sin 120575cos3120575]

+ 119903Moon120585 [(

1198912

2

) 1205752 csc 120575 + (

1198912

2

) sin 120575cos2120575])

(7)



1cos 1205762) cos 120582)

sdot ((sin 120582 cos 1205762)2

+ [sin (1205932minus 1205931)

+2 cos 1205762sin 1205761sin2 (120582

2

)]

2

)

minus12

)

120585 = 1 minus (

(cos 1205761cos 1205762) sin 120582

sin 120575

)

2

(8)

where

120582 = 120582 (119905imp) minus 120582 (119905119894)


119903pol sin120601 (119905119894)

119903Moon cos120601 (119905119894)

)

1205762= tanminus1(

119903pol sin120601 (119905imp)

119903Moon cos120601 (119905imp))

1205932minus 1205931= (120601 (119905imp) minus 120601 (119905

119894))

+ 2 [sin (120601 (119905imp) minus 120601 (119905119894))]


2+ 1205783) 119903pol]

120578 = (


119903Moon + 119903pol)

(9)






(10)





RLVLHSC2C = [Q

1]119879RSC2C (11)

VLVLHSC2C = [Q

1]119879VSC2C (12)

RLVLHC2SC = [Q

2]119879RC2SC (13)

VLVLHC2SC = [Q

2]119879VC2SC (14)




Q2=

[[[[[[[[[[

[

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

]]]]]]]]]]

]

(15)





SC2C

10038161003816100381610038161003816RLVLHSC2C

10038161003816100381610038161003816)119895LVLHSC

(RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816)119894LVLHSC

)

120573 (119905)SC2C = sinminus1((

RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816

)

119896LVLHSC

)


C2SC

10038161003816100381610038161003816RLVLHC2SC

10038161003816100381610038161003816)119895LVLHCube

(RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816)119894LVLHCube

)

120573 (119905)C2SC = sinminus1((

RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816

)

119896LVLHCube

)

(16)







Mother-ship

Mother-shiporbit

Impact trajectory

Impact point

Moon center

CubeSat

119829SC

119829Cube

LVLHSC

LVLHSC

LVLHSC

120573(t)SC2C

120572(t)SC2C

119825SC

120573(t)C2SC

120572(t)C2SC

LVLHCube

LVLHCube

LVLHCube

119825Cube


SC2C





90100

110120

130140

150160

170180

010203040

50607080

90

0

20

40

60

80

100

Clos

est a

ppro

ach

altit

ude (

km)

Impact region

Nonimpact region








90 More than 90 NA100 6750 9000110 5150 9000120 4100 9000130 3400 9000140 2950 9000150 2700 9000160 2500 9000170 2400 9000180 2350 9000










015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)


(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)


-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)


(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)






minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894)



119889 (119905119894)

= 119903pol ([(1 + 119891 + 1198912) 120575]

+ 119903Moon [[(119891 + 1198912) sin 120575 minus (

1198912

2

) 1205752 csc 120575]]

+ 120585 [minus(

119891 + 1198912

2

) 120575 minus (

119891 + 1198912

2

)

sdot sin 120575 cos 120575 + (

1198912

2

) 1205752 cot 120575]

+ 1199032

Moon [minus(

1198912

2

) sin 120575 cos 120575]

+ 1205852[(

1198912

16

) 120575 + (

1198912

16

) sin 120575 cos 120575

minus(

1198912

2

) 1205752 cot 120575 minus (

1198912

8

) sin 120575cos3120575]

+ 119903Moon120585 [(

1198912

2

) 1205752 csc 120575 + (

1198912

2

) sin 120575cos2120575])

(7)



1cos 1205762) cos 120582)

sdot ((sin 120582 cos 1205762)2

+ [sin (1205932minus 1205931)

+2 cos 1205762sin 1205761sin2 (120582

2

)]

2

)

minus12

)

120585 = 1 minus (

(cos 1205761cos 1205762) sin 120582

sin 120575

)

2

(8)

where

120582 = 120582 (119905imp) minus 120582 (119905119894)


119903pol sin120601 (119905119894)

119903Moon cos120601 (119905119894)

)

1205762= tanminus1(

119903pol sin120601 (119905imp)

119903Moon cos120601 (119905imp))

1205932minus 1205931= (120601 (119905imp) minus 120601 (119905

119894))

+ 2 [sin (120601 (119905imp) minus 120601 (119905119894))]


2+ 1205783) 119903pol]

120578 = (


119903Moon + 119903pol)

(9)






(10)





RLVLHSC2C = [Q

1]119879RSC2C (11)

VLVLHSC2C = [Q

1]119879VSC2C (12)

RLVLHC2SC = [Q

2]119879RC2SC (13)

VLVLHC2SC = [Q

2]119879VC2SC (14)




Q2=

[[[[[[[[[[

[

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

]]]]]]]]]]

]

(15)





SC2C

10038161003816100381610038161003816RLVLHSC2C

10038161003816100381610038161003816)119895LVLHSC

(RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816)119894LVLHSC

)

120573 (119905)SC2C = sinminus1((

RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816

)

119896LVLHSC

)


C2SC

10038161003816100381610038161003816RLVLHC2SC

10038161003816100381610038161003816)119895LVLHCube

(RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816)119894LVLHCube

)

120573 (119905)C2SC = sinminus1((

RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816

)

119896LVLHCube

)

(16)







Mother-ship

Mother-shiporbit

Impact trajectory

Impact point

Moon center

CubeSat

119829SC

119829Cube

LVLHSC

LVLHSC

LVLHSC

120573(t)SC2C

120572(t)SC2C

119825SC

120573(t)C2SC

120572(t)C2SC

LVLHCube

LVLHCube

LVLHCube

119825Cube


SC2C





90100

110120

130140

150160

170180

010203040

50607080

90

0

20

40

60

80

100

Clos

est a

ppro

ach

altit

ude (

km)

Impact region

Nonimpact region








90 More than 90 NA100 6750 9000110 5150 9000120 4100 9000130 3400 9000140 2950 9000150 2700 9000160 2500 9000170 2400 9000180 2350 9000










015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)


(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)


-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)


(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)






minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Q2=

[[[[[[[[[[

[

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119909Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119910Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

(


1003816100381610038161003816

)

119911Cube

]]]]]]]]]]

]

(15)





SC2C

10038161003816100381610038161003816RLVLHSC2C

10038161003816100381610038161003816)119895LVLHSC

(RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816)119894LVLHSC

)

120573 (119905)SC2C = sinminus1((

RLVLHSC2C

1003816100381610038161003816RLVLHSC2C

1003816100381610038161003816

)

119896LVLHSC

)


C2SC

10038161003816100381610038161003816RLVLHC2SC

10038161003816100381610038161003816)119895LVLHCube

(RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816)119894LVLHCube

)

120573 (119905)C2SC = sinminus1((

RLVLHC2SC

1003816100381610038161003816RLVLHC2SC

1003816100381610038161003816

)

119896LVLHCube

)

(16)







Mother-ship

Mother-shiporbit

Impact trajectory

Impact point

Moon center

CubeSat

119829SC

119829Cube

LVLHSC

LVLHSC

LVLHSC

120573(t)SC2C

120572(t)SC2C

119825SC

120573(t)C2SC

120572(t)C2SC

LVLHCube

LVLHCube

LVLHCube

119825Cube


SC2C





90100

110120

130140

150160

170180

010203040

50607080

90

0

20

40

60

80

100

Clos

est a

ppro

ach

altit

ude (

km)

Impact region

Nonimpact region








90 More than 90 NA100 6750 9000110 5150 9000120 4100 9000130 3400 9000140 2950 9000150 2700 9000160 2500 9000170 2400 9000180 2350 9000










015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)


(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)


-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)


(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)






minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Mother-ship

Mother-shiporbit

Impact trajectory

Impact point

Moon center

CubeSat

119829SC

119829Cube

LVLHSC

LVLHSC

LVLHSC

120573(t)SC2C

120572(t)SC2C

119825SC

120573(t)C2SC

120572(t)C2SC

LVLHCube

LVLHCube

LVLHCube

119825Cube


SC2C





90100

110120

130140

150160

170180

010203040

50607080

90

0

20

40

60

80

100

Clos

est a

ppro

ach

altit

ude (

km)

Impact region

Nonimpact region








90 More than 90 NA100 6750 9000110 5150 9000120 4100 9000130 3400 9000140 2950 9000150 2700 9000160 2500 9000170 2400 9000180 2350 9000










015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)


(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)


-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)


(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)






minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














90 More than 90 NA100 6750 9000110 5150 9000120 4100 9000130 3400 9000140 2950 9000150 2700 9000160 2500 9000170 2400 9000180 2350 9000










015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)


(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)


-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)


(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)






minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










015

3045

6075

90

90105

120135

150165

1800

50

100

Cube

Sat fl

ight

time (

min

)


(a)

015

3045

6075

90

90105

120135

150165

1800

5000

10000

Cros

s ran

gedi

stan

ce (k

m)


-V(m

s)

(b)

015

3045

6075

90

90105

120135

150165

1800

2

4

Impa

ct an

gle

(deg

)


(ms)

(c)

015

3045

6075

90

90105

120135

150165

1801600

1700

1800

Impa

ct v

eloc

ity(m

s)


(ms)

(d)






minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus1

minus05

0

05

1


minus10

1

X (LU)Y (LU)

Z(L

U)



at impact

(a)

minus1

minus05

0

05

1

minus1 0 1

minus10

1

X (LU)Y (LU)

Z(L

U)



(b)

minus1

minus05

0

05

1


minus1

0

1

X (LU)Y (LU)

Z(L

U)



at impact

(c)






Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Releasepoint







ground track





minus80

minus60

minus40

minus20

0

20

40

60

80

100

Longitude (deg)

Are

odet

ic la

titud

e (de

g)






















Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Case A

Case B

Case C


02

04

06

08

1

Nor

mal

ized

areo

detic

altit

ude

(a)



Case ACase B

Case C0

1

2

3

4

5

Impa

ct an

gle (

deg)

(b)





0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case A


(a)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


Case A

(b)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

Case B


(c)

0

50

100O

ut-o

f-pla

ne d

irect

ion

(deg

)

minus100

minus50


Case B

(d)

0

50

100

150

200


In-p

lane

dire

ctio

n (d

eg)

C to MM to C

Case C


(e)

0

50

100

Out

-of-p

lane

dire

ctio

n (d

eg)

minus100

minus50


C to MM to C


Case C

(f)



00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

50

100

150

200

250


Relat

ive d

istan

ce (k

m)

Case ACase BCase C


(a)

0

50

100

150

200

250

Relat

ive v

eloc

ity (m

s)


Case ACase BCase C


(b)





5 Conclusions






Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Lunar CubeSat Impact Trajectory...

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Transcript of Research Article Lunar CubeSat Impact Trajectory...