IAA-AAS-DyCoSS-14-08-09 FORMATION FLYING DYNAMICS OF MICRO- SATELLITES NEAR EQUATORIAL LOW ORBITS UNDER THE INFLUENCE OF J AND J Haijono Djojodihadjo 1 The pesent wok assess the effet of the Eath s oblateness paametes, patiulaly J and J, on the fomation flight of mio satellites in nea equatoial low obits. The modified Hill-Clohessy-Wiltshie equations ae extended to inlude the J and J effets in the LVLH fame of efeene. The pimay gavitational petubation effet is due to the equatoial bulge tem, J. The J tem hanges the obit peiod, a dift in peigee, a nodal peession ate and peiodi vaiations in all the obital elements. The J zonal hamoni aptues the equatoial bulge of the Eath, and is the lagest oeffiient desibing the Eath s shape. Thee is about a 1km diffeene in equatoial and pola adii due mainly to this bulge. In the Eath obit about 800 km altitude, the J effet is muh lage in ompaison with othe petubations suh as atmosphei dag, sola adiation pessue and eletomagneti effets. The modified lineaized Hill-Clohessy-Wiltshie to take into aount the influene of J and J is utilized to detemine the obits of twin spaeaft in fomation flight in Nea Equatoial obits, whee the vaiation of J is less appaent. A simplified appoah, apitalizing on the balane between lineaized appoah and expeted fidelity of the obtained solution, has been synthesized to aive at a lineaized J and J modified HCW equation. The omputational esults obtained ae assessed by ompaison to Shweighat-Sedwik fomula. The signifiane and elevane of the influene of these paametes in the detemination and design of fomation flying obits ae assessed though paameti study. As a patiula example, fo low eath obit (i.e. 847 km), the eo is about 0.5km fom the desied elative position in the LVLH o Hill fame afte 16.67 hous. Futhe ompaison to simila esults in the liteatue exhibits the plausibility of the wok. Key wods : Fomation Flying, Gavitational Potential, J, J, Nea-Equatoial Low Obits, Obital Mehanis INTRODUCTION Reseah on autonomous satellite fomation flying has been of geat inteest and need in the last few deades, and multiple spaeaft to eplae a single lage satellite will be an enabling tehnology fo many missions to ome. These lustes of spaeafts usually wok togethe to aomplish a mission, and ae of geat inteest fo intefeomety, spae-based ommuniations, and missions to study the magnetosphee, to ite a few examples. The benefits to using multiple spaeaft inlude ineased podutivity, edued mission and 1 Pofesso, Aeospae Engineeing Depatment, Faulty of Engineeing, Univesiti Puta Malaysia haijono@djojodihadjo.om 1
launh osts, gaeful degadation, on-obit eonfiguation options, and the ability to ay out missions that would not be possible othewise. The need to auately detemine and ontol the position of satellites within the fomation have elied on the Hill s equations, also known as the Clohessy-Wiltshie equations, due to thei lineaity and simpliity, and is efeed hee as the Hill-Clohessy-Wiltshie s (HCW) equation. Hill-Clohessy-Wiltshie s lineaized equations of elative motion ae a set of lineaized equations that desibe the elative motion of two spaeaft in simila nea-iula obits assuming Kepleian ental foe motion. With the need fo bette fidelity in the esults of the lineaized appoah, in patiula to take into aount elevant distubanes in low Eath obits suh as the influene of the Eaths s oblateness, atmosphei distubanes and the like, thee is a need to modify the Hill- Clohessy-Wiltshie s lineaized equations. Many eseahes have intodued modifiations to aptue the effet of the dominant Eaths s oblateness paamete on the fomation flying satellite luste. The Eath gavitational potential an be epesented by spheoidal hamonis (Refeenes 1,, and 4). Lineaizing the gavitational tems in the pesene of the dominant oblateness paamete, J, fo the deputy satellite with espet to the hief s efeene obit, analytial solutions simila to that of the HCW equations an be deived. These J Lineaized Modified Hill s Equations desibe the mean motion hanges in both the in-plane and out-of-plane motion, taking into aount the lineaization assumptions (Refeenes 5, 6, 7 and 8). A new set of onstant oeffiient, lineaized diffeential equations of motion an be obtained, whih ae simila in fom to Hill-Clohessy-Wiltshie s equations, but they aptue the effets of the J distubane foe. Othe eseahe (e.g. Refeene 9) has also intodued modifiations to the HCW equations in simila fashion to inlude the effets of atmosphei dag while maintaining thei lineaity and simpliity. With the gowing tend towad eduing the size of satellites, aeful modeling of the petubation foes on fomation flying of Mio-Satellites in low obits is neessay. Then the objetive of the pesent wok is the investigation of Eath s oblateness effets on the fomation flight of mio satellites in low iula obits and the assessment of thei influene on fomation maintenane. In ode to study spaeaft fomation flight in low Eath obits, in addition to the effet of J, the pesent pape will also investigate the additional influene of the next tem J of the gavity potential. Fo this pupose, the Hill-Clohessy-Wiltshie s equations whih have been modified by Shweighat and Sedwik (Refeenes 5 and 6) as a set of lineaized diffeential equations of motion to inlude the J effets will be extended to inopoate the effets of J fo iula obits while maintaining thei lineaity and simpliity. In addition to low iula obits, the extension will inlude efeene obits of small eentiity. Patiula attention is given to nea equatoial obits, whee J an be onsideed to be onstant. Numeial simulation esults based on suh model will be pesented to eveal the effets of eah of the two dominant tems haateizing the Eath Oblateness Effets on the fomation flight of mio satellites in low nea equatoial iula obits. Paameti studies ae aied out with espet to obital inlination and adius, and hief-deputy satellites sepaation distane. Results obtained will be disussed with efeene to the influene of othe distubing foes that have been pesented in the liteatue. The influene of the oblateness paamete J hanges the natue of the obit of the Chief satellite taken as the efeene obit than that appoximated by the Hill-Clohessy-Wiltshie equation, whih has been metiulously elaboated by Shweiget and Shweiget and Sedwik. The edeivation of the satellite elative motion inopoate oetion of the obital peiod of the efeene obit, the nodal dift of the efeene obit and the oss-tak motion, taking into aount the initial onditions. The pesent wok intodued futhe simplifiations by onsideing nea equatoial obits. Closed fom solutions fo the influene of oblateness paamete has been woked out by Gufil (efeenes 1 and 14).
COORDINATE SYSTEM TRANSFORMATION Refeing to Coodinate System onventionally utilized (Refeenes 4, 10 and 11), hee the subsipt N denotes a veto in the ECI fame, and a subsipt O denotes a veto in the satellite-enteed fame. The ( θ i ) oodinate system (o Eath Centeed Chief Satellite Obital Plane oodinate system) is used in desibing the J distubane in the loal ( x - y - z) oodinate system. The two Eule angles, θ and i, and omplete the assoiated geometial tansfomation fom the ECI fame to the ( θ i ) fame, whee the dietion osine matix is fomed by the -1- Eule angle sets Ω, i and θ. Figue 1. The ( θ i ) oodinate system is used in desibing the J distubane in the loal ( x - y - z ) oodinate system. This is depited in Figue 1 and defined as the longitude of the asending node, the agument of latitude, and the inlination angle, espetively, and is given by (Refeene 7): osos sin sin osi sin os os sin os i sin sin i ON os sin sin os osi sin sin os os osi os sin i (1a) sin sini ossini osi o ex ey ez ON e X e Y e Z (1b) e hx ehy e hz BASELINE HILL-CLOHESSY-WILTSHIRE EQUATION Baseline Hill-Clohessy Wiltshie Equations, fo iula obit aound the Eath as the ental body, assumed the Eath as point mass enteed at its ente of mass and the ente of the obit. The Hill-Clohessy Wiltshie Equations of motion in the hief LVLH fame ae given by. d x dy x 0 (a) dt dt d y dx 0 (b) dt dt d z z 0 () dt whih is also known as the Unpetubed HCW Equations. The angula veloity is given by
m G M The out-of-plane motion is modeled as a hamoni osillato, whee the in-plane motion is desibed as oupled hamoni osillatos. These seond-ode diffeential equations have the geneal solutions x t Aos nt x off (4a) y t A sinnt nx t y (4b) os off off z t B nt (4) whee A, α, x off, y off,b and β ae the six integal onstants. The veloities ae found as the time deivatives of (6a,b,). In ode to podue bounded elative motion, the adial offset tem must be equal to zeo to eliminate the seula gowth pesent in the alongtak dietion. Fo the z dietion, integation of: z t B 0 sin t (5) B0 yields zt ost D0 o zt Bosnt D0 (6) Following Refeene 11, the analytial solutions of the homogeneous CW equations ae obtained as follows. Define X x y z T and x y z T V. A subsipt 0 denotes the initial ondition. Then the solution of the lineaized Clohessy-Wiltshie (CW) equations an be epesented in the following matix fom: Xt XX txt0 XV tv t0 (7a) Vt txt tv t (7b) VX 0 VV 0 whee XX (t), XV (t), VX (t) and VV (t) ae state-tansition maties defined in Refeene 4 and elaboated in Refeene 11. The homogeneous solutions of the CW equation detemine the position and the veloity of the deputy spaeaft elative to the hief spaeaft as a funtion of t subjet to initial onditions X 0 and V 0. RELATIVE BOUNDED MOTION In fomation flying, the motion of deputy satellite must emain bounded with espet to the hief satellite suh that it expeienes no seula dift and the fomation onfiguation is maintained. One needs to find the ondition suh that the solutions of the Clohessy-Wiltshie equations ae bounded (Refeene 10). Equation (a) and Equation (b) ae oupled and they an be solved in paallel. Equation (b) an be ewitten fo y (t) : 0 0 y t x t x y (8) If one integate Equation (8) fom 0 to t, one find tems that gow unboundedly ove time, namely the tems x0 t and y0 t. Howeve, yt an be made bounded and peiodi given the ondition x0 y 0 0 (9) Then, the solution fo the in plane motion of the deputy satellite is: x t A sin 0 t (10a) os y t A t C (10b) 0 0 whee A 0, phase angle α and integation onstant C 0 depend on the initial onditions. The outof-plane motion is deoupled fom the in-plane motion and its solution takes on the fom of a simple hamoni osillato: z t B 0 sin t (10) whee amplitude B 0 and phase angle ae onstants whih depend on the initial onditions. The out-of-pane motion is peiodi and bounded with espet to the hief satellite. The set of solutions in Equation (10) define a family of bounded, peiodi motion tajetoies fo the deputy satellite in the elative fame unde the assumptions of the HCW-equations. The () 4
motion of the deputy satellite, if pojeted onto the y-z plane, follows an ellipse of semi-majo axis A 0 and semi-mino axis A 0. J GRAVITATIONAL PERTURBATION EFFECTS The pevious setion assumed the ental body was a sphee of unifom density. This allows the two-body equations of motion to be witten in a moe simplified fom. Howeve, the Eath is not a pefet sphee with unifom density. Theefoe, we would like to detemine the gavitational potential due to an aspheial ental body. In ode to detemine the gavitational potential at point P, eah point in the Eath, Q m must be taken into aount. The angles φ sat and φ Q ae the espetive olatitudes, λ Q and θ sat ae the longitudinal aguments, and Λ is the angle between the vetos Q and sat, also known as the gound ange o total ange angle. All the above angle measuements ae geoenti. The potential that desibes an aspheial ental body is then given (Refeenes 1, and ) as: l l l R R U 1 Jl Pl osg P, os, os, sin sat l m g C sat l m msat Sl m msat l l m1 (11) whee J l, C l,m, and, S l,m ae gavitational oeffiients and R is the equatoial adius of the Eath. The gavitational potential is 14 GM e.986005 10 m / s (1) whih is the fist tem of the moe geneal Eath s gavitational potential. The fist tem is the two-body potential, wheeas the seond tem is the potential due to zonal hamonis ( J ll tems, whee m=0, and epesent bands of latitude). An aspheial body whih only deviates fom a pefet sphee due to zonal hamonis is axially symmeti about the Z-axis. The thid tem epesents two othe hamonis. The setoial hamonis, whee l = m, epesent bands of longitude, and tesseal hamonis, whee l m 0, epesent tile-like egions of the Eath. The J oeffiient is about 1000 times lage than the next lagest aspheial oeffiient, and is theefoe vey impotant when desibing the motion of a satellite aound the Eath. If the gavitational potential is onsideed to be due to spheial Eath, i.e. the fist tem of equation (11), then the gavitational aeleation is given by g (1) If J is inluded, the gavitational potential due to the J distubane an be obtained (fom Refeene 1, and 4) as R Uzonal J P osg sat whih an futhe be edued to Re J 1 U os is the assoiated Legende polynomial of J and the seond zonal gavitational oeffiient aoding to the JGM- model has been alulated as J = 1.08669568815 10. Similaly J =.556 10 6. The aeleation due to J in the ECI fame is then alulated as the gadient of the potential whee P os gsat 1 sin i sin JR U J J sin isinos 4 siniosisin (1) (14) (15) 5
The hief and deputy equations of motion an be ewitten in the inetial fame as J (16) d d Jd (17) d The aeleation due to J in the LVLH fame may be alulated fom the gadient in the and Z dietions: U J U J 15 z z U J e ez JR e 4 6 e 5 z (18) z The ( θ i ) oodinate system (Eath Centeed Chief Satellite Obital Plane oodinate system) is used in desibing the J distubane in the loal ( x - y - z ) oodinate system. The pesene of and the two Eule angles, θ and i, omplete the geomety of the assoiated tansfom fom the ECI fame to the ( θ i ) fame, utilizing the dietion osine matix fomed by the -1- Eule angle sets Ω, i and θ. This is defined as the longitude of asending node, the agument of latitude, and the inlination angle, espetively. x osos sin sin osi sin os os sin osi sin sini I X y os sin sin os osi sin sin os os osi os sini IY (19) z sin sini ossini osi I Z The aeleation due to J in the LVLH fame may be alulated fom the gadient in the and Z dietions: U J U J 15 z z U J e ez JR e 4 6 e 5 z (0) z whee the Z omponent may be expessed in the LVLH fame as: e sin sin sin os os z i e i e ie (1) z os sinisin () Substituting this bak into Eq.(0) yields the aeleation due to J gadient to be JR 1 sin isin ˆ U ˆ ˆ sin sin os sin os sin J J 4 i i j i i k (a) o 1 sin i sin sin i sin sin i sin 6JR 1 1 7 sin i os J sin sin sin sin 5 i i 4 4 sin i os 1 5 sin isin sin i sin 4 4 4 (b) in Eath-Centeed Inetial (ECI)fame of efeene. The effets of J ae intinsially detemined fom the aspheial ental body. J MODIFIED HCW EQUATION The hief and deputy equations of motion in the inetial fame due to J in the ECI fame is given by equations (16) and (19) whih appea as two-body Kepleian motion with added J petubations and is the exat o tuth model.. Simila to unpetubed HCW ase, in LVLH, the solution of the equations of motion an be epesented in the following matix fom, whee appopiate tems have to be fomulated: 6
Xt XX X0 XV V 0 (4) and Vt VX X0 VV V 0 (5) With the pesent baseline fomulation, the appoah follows losely the lineaized appoah of Shweighat's (Refeene 5) and Ginn's (Refeene 7). Computational poedue and ode is then developed. The inetial elative position and veloity is defined as the position and veloity of the deputy elative to the hief. d N (7) d (8) N The elative position in the LVLH is alulated using the dietion osine matix,[on], defined by Eq. (1a) and Eq. (1b). Hene: ON O (9) N The elative veloity in the LVLH fame is defined using the tanspot theoem as ON / (0) O N O whee ON / is the otation ate of the LVLH fame, whih is the angula veloity of the hief obit as stated befoe, and is defined as the angula momentum times the magnitude of the position squaed: ON / (1) As the elative position and veloity is now fully defined fom the exat nonlinea equations of motion in the ECI fame (Eqns. (7-1), then these ae the govening exat model to be used as a basis fo ompaison. The Equation of motion fo lineaized J modified HCW equation beomes (efeenes 6 and 7): d x dy R 1 1 os i n 5 n x n J sin isin kt dt dt 8 (a) d y dx R n n J sin i sin kt (b) dt dt whee d z n z 0 dt () JR (d) k n os i 7 The equations fo the alulation of the influene of J on the lineaized (HCW) obit ae summaized below 5 1 1 AJ k n 1 s sin i s s xt xh0 y h0 os n 1 st s 1 n s 1 4 k n n s 4k (a) 1 AJ k n 1s sin i x 4 0 s 1 h 1 s os kt sin n 1 st xh0 yh0 4 k n n s 4k n 1 s s 1 n s 1 7
5 1 41 1 AJ ns k 1 s n sin i s s s y t x h0 y h0 sin n 1 st 1 s n1 s k 1 s n n s 4k whee AJ 5n s 4k n 6nk 1 s sin i s 1 1 8 k n n s k 1 sin kt x h0 os n 1 st 4 n s z h 0 h0 os 1 sin 1 z t z n st n st n 1 s (b) () R AJ n J (d). These equations ae inopoated in the MATLAB TM omputational outine following the sheme depited in Figue. METHOD OF PERTURBATIONS The following deivations ae adapted fom Refeenes 7 and 1, and add the effets of seond-ode diffeential gavity to the HCW equations using the method of petubations, assuming that the J lineaized modified HCW Eqns. (4.0) have solutions of the following fom: x x x x x x x x x (4a,b,) h p h p h p y y y y y y y y y (4e,f,g) h p h p h p z z z z z z z z z (4h,i,j) h p h p h p The subsipt h efes to the solutions of the HCW equations and in this ontext will be temed the homogeneous solution; the subsipt p efes to the oetion due to the nonlinea gavitational tems and will be temed the petubation solution. Define ou petubation paamete as (5) 4 It is noted that this petubation paamete is not dimensionless fo ompaison with Refeenes 5 and 6. The new nonlinea equations of motion ae now x ny n x y z x (6a) y nx xy (6b) z n z xz (6) Substituting Equation (4.1) into (4.) we obtain xh ny h n xh xp ny p n xp yh yp zh z p xh xp (7a) y nx y nx x x y y (7b) h h p p h p h p z n z z n z x x z z (7) h h p p h p h p Sine ε<<1, afte seies expansion and then dopping the highe ode tems of ε, the J lineaized modified HCW equations of motion fo the seond-ode gavitational petubation in petubed fom by Ginn beomes: x ny n x y z x (8a) h h h h h h y nx x y (8b) h h h h h h h h z n z x z (8) 8
The patiula the solution to the HCW equations may be solved fo by diet integation yielding an analytial model. The effets of seond-ode diffeential gavity may be added to the J -Modified HCW equations using the method of petubations x x x (9) h whee the homogeneous solution is the exat solution obtained fom the J Modified HCW equations, and the petubed solution is solved by the following linea, onstant oeffiient, diffeential equations: d xp dy p n 0 n0 x yh zh xh (40a) dt dt d yp dxp n 0 xhyh (40b) dt dt d zp n 0 z p xhzh (40) dt J and J MODIFIED HCW EQUATION Simila to the deivation of J modified lineaized HCW equation, the influene of J and J an be deived following the development below J J (41a) d d d d d p J J (41b) The aeleation due to J in the LVLH fame may be assumed to follow simila deivation fo J hene it an be alulated fom the gadient in the and Z dietions: UJ UJ U J e ez (4a) z 1 sin isin e x JR U J J sin isinos e 4 y (4b) siniosisin e z These equations may be dietly integated to obtain the losed-fom solution of the petubation. When ompaed to the two diffeent solutions that only take one of the distubanes into aount, it is seen that this new ombined solution yields bette esults when ompaed to the tuth model, poviding fo peiodially bounded solutions in all elative omponent dietions. It was expeted that the addition of the J and J petubation to the HCW equations would be to edue the gowth in the oss-tak dietion. It was also expeted that adding the petubation assoiated with seond-ode diffeential gavity would impove upon the in-plane motion of the new solution, and is done so by bounding the eos fom tuth in the along-tak dietion. The fidelity of this new petubed J - Modified HCW model may be impoved upon to eliminate the unbounded gowths in amplitude in the adial and oss-tak dietion, by taking into aount the peession of the angula veloity veto. It is also believed that futhe analysis by adding this influene will povide fo a stonge iteion fo eliminating the seula gowth in the along-tak dietion. COMPARISON OF BASELINE CLOHESSY-WILTSHIRE MODEL OF TWIN- SATELLITE ORBITS WITH J-PERTURBED ONES To demonstate the influene of J on the lineaized (HCW) obit of the Twin Satellite Fomation Flying Obits, the J petubed lineaized HCW equations obits ae ompaed with 9
the baseline ones. The initial ondition ae those given in Table 1. The esults ae exhibited in Figs. to 7.. Figue : Compaison of the X,Y and Z values, espetively, of Deputy Satellite obit aound the Chief Satellite as the solution between baseline HCW, linealy J modified HCW Equation and Shweighat s esults (the latte two inopoate the effet of J ). Table 1. CHIEF'S ORBITAL ELEMENTS AND DEPUTY'S INITIAL CONDITIONS WITH RESPECT TO CHIEF'S Chief Satellite Altitude, h (km) 847 Eentiity, e 0 Obit Inlination, I (deg) 10 0 Right Asension of the Asending Node, Ω (deg) 0 0 Agument of Peigee ω (deg) 0 0 Mean Anomaly at Epoh, M (deg) 0 0 Deputy Satellite Stating Condition (Chief-enteed Fame) x 0 (km) 0.0 y 0 (km) 5.0 z 0 (km) 0.0 v x0 (km/s) 0.5785 10 - v y0 (km/s) 0.0 v z0 (km/s) 1.1570 10 - i (deg) 10 θ(deg) nt 10
Figue. Compaison of the Deputy Satellite obital adius aound the Chief Satellite as the solution of Clohessy-Wiltshie Equation (without J ) and inopoating the influene of J, using lineaized modified Clohessy-Wiltshie Equation. Figue 4. The diffeene between the adius of the obit of the Deputy Satellite aound the Chief Satellite as the solution of the oiginal lineaized HCW Equation (without J ) and that inopoating the influene of J, using lineaized J modified HCW Equation Figue 5. Compaison of Baseline Gound-Tak of the Deputy and Chief Satellites obits as the solution of baseline HCW and the Linealy Modified HCW equation whih inopoate the influene of J. 11
Figue 6. Compaison of Deputy Satellite obit aound the Chief Satellite as the solution of baseline Clohessy-Wiltshie Equations (without J ) and Linealy Modified HCW equations whih inopoate the influene of J. Figue 7: Compaison of the Deputy Satellite obital adius aound the Chief Satellite as the solution of the baseline Clohessy-Wiltshie Equations, the lineaized J modified HCW Equations and simila solution obtained by Shweighat [17]. The esults show that the equations deived in this wok have lose similaity with the ones deived by Shweighat, although quantitatively thee ae diffeenes. It should be noted that Shweighat s solutions oiginate fom diffeent J lineaization ompaed to the pesent wok. Suh diffeene may be attibuted to the notion that the peset wok does not inlude the dift of the asending node of a satellite unde the influene of the J distubane. COMPARISON OF BASELINE CLOHESSY-WILTSHIRE MODEL OF TWIN- SATELLITE ORBITS WITH J AND J-PERTURBED ONES Simila to the deivation of the J Lineaized Modified HCW Equations, the influene of J to the latte equation is deived using petubation appoximation. J, whih is.556 10 6, is about 1000 times smalle than J, podues pea-shaped vaiations: a ~17 m bulge at Noth pole and ~7 m bulges at mid-southen latitudes. The detail is elaboated in a ompanion pape (Djojodihadjo, Refeene 1). 1
U JR 5Z Z JR 5Z Z U J 5 7 5 (4) 1 J z J J 5z z u sin 4 5sin 7 5z 7 5 (44) JR 5Z Z JR 5 Z Z U J should be U 7 5 J (45) J R 5 5 P os J R os os J R z z sat sat sat J g g g UJ UJ U J e ez z JR 15Z 5Z 15Z e 6 8 e 7 5 z (46) In the following figues, the omputational esults obtained using J Lineaized Modified HCW Equations ae ompaed to the baseline HCW as well as J Lineaized Modified HCW Equations esults. Figue 8 The diffeene in the adius of the obit of the Deputy Satellite aound the Chief Satellite between the solution of JLMCW and the solution of JLMCW. Figue 9. Compaison of the adius of the obit of the Deputy Satellite aound the Chief Satellite as the solution of CW, JLMCW and JLMCW. 1
Figue 10. Compaison of the diffeene in the distane of the path of Deputy Satellite aound the Chief Satellite in LVLH oodinate between the solution of CW Equation with JLMCW equation and CW Equation with JLMCW equation CONCLUSION Lineaized Hill-Clohessy-Wiltshie equations have been utilized in developing modified fom to take into aount the influene of J on the obits of twin spaeaft in fomation flight in nea-eath obits. Fo Nea Equatoial obits the vaiation of J is less appaent. Vaious elevant appoahes and eent wok on this issue have been synthesized into a novel and simplified appoah, apitalizing on the balane between lineaized appoah and expeted fidelity of the obtained solution, as stipulated by many ealie wok. Judging fom the auay estimation of simplified lineaized appoah, the exhibited omputational esults wee obtained using J lineaized HCW equation. The oiginal (baseline) lineaized HCW appoah and lineaized J -modified HCW equation also exhibit the meit of simple analysis, whih ould be extended to inopoate othe paametes. Simila analysis has been also aied out to obtain the influene of J on the fomation flying of spaeafts in Nea Equatoial Low Eath obits. The elevane of paameti study as a peliminay step towads optimization effots has been demonstated in the pesentation of the esults. The omputation that has been pefomed using in-house developed MATLAB pogam. As a patiula example, fo nea equatoial low eath obit (i.e. 847 km), the eo is about 0.5km fom the desied elative position in the LVLH o Hill fame afte 16.67 hous. Aknowledgment The autho would like to thank Univesiti Puta Malaysia (UPM) and Ministy of Highe Eduation, Malaysia fo ganting Reseah Univesity Gant Sheme (RUGS) Pojet Code: 97800 and Exploatoy Reseah Gant Sheme (ERGS) ERGS 557088, espetively, unde whih the pesent eseah is aied out. Refeenes 1 Vinti, John P., (1971) Repesentation of the Eath's Gavitational Potential, MIT Measuement System Lab Repot RE-71-NACA CR-11589. Djojodihadjo, Haijono (1974.), Vinti's Sufae Density As A Means Of Repesenting The Eath's Distubane Potential, Poeedings ITB (ITB Jounal), Vol.7, No. 4. Andeson, D.L. (1989), Theoy of the Eath, Ch.1, Blakwell Si.Publ. 4 Alfiend, Kyle T., Shaub, Hanspete, Gim, Dong-Woo (000), Gavitational Petubations, Nonlineaity and Ciula Obit Assumption Effets on Fomation Flying Contol Stategies, AAS 00-01 5 Shweighat, S. A., Development and Analysis of a High Fidelity Lineaized J Model fo Satellite Fomation Flying, M.S. thesis, Massahusetts Institute of Tehnology, 001 14
6 Shweighat and Sedwik Development and Analysis of a High Fidelity Lineaized J Model fo Satellite Fomation Flying-AIAA 001-4744 7 Ginn, J.S. (006), Spaeaft Fomation Flight-Analysis of the Petubed J-Modified HCW Equation, MS Thesis, Univesity of Texas Alington. 8 Jennife A Robets, Satellite Fomation Flying Fo an Intefeomety Mission, PhD Thesis, Canfield Institute of Tehnology, 005 9 Reid, T.; Misa, A.(011), Fomation flight of satellites in the pesene of Atmosphei Dag, Jounal of Aeospae Engineeing, Sienes and Appliations, Jan Apil 011, Vol. III, No 1 65. 10 Alfiend K.T., Vadali S.R., Gufil P., How J.P., Bege L.S. (010), Spaeaft Fomation Flying: Dynamis, Contol and Navigation, Elsevie Astodynamis Seies, Bulington, USA. 11 Vaddi, S.S., Vadali, S.R., and Alfiend, K.T., Fomation Flying: Aommodating Nonlineaity and Eentiity Petubations, Jounal of Guidane, Contol, and Dynamis, Vol. 6, No., 00, pp. 14-. 1 Djojodihadjo, Haijono and Haithuddin, A.Salahuddin M.( 010), Spaeaft Fomation Flying Fo Topial Resoues And Envionmental Monitoing: A Paameti Study, AAS 10-441. 1 Djojodihadjo, Haijono, (014), Futhe Development of The Influene of the Eath s Gavitational Potential Petubation on the Fomation Flying Obits of Mio-Satellites Nea Equatoial Low Obits, Intenal Repot, Univesity Puta Malaysia, Aeospae Engineeing Depatment. 14 Laa, M and Gufil, P., Integable appoximation of J-petubed elative obits, Celest Meh Dyn Ast (01) 114:9 54 15 Matinusi, V. and Gufil, P., Solutions and peiodiity of satellite elative motion unde even zonal hamonis petubations 15