Nonlinear Orbital Dynamic Equations and State- Dependent Riccati Equation Control of Formation Flying Satellites 1

Transcription

1 Noninear Orbita Dynamic Equations and State- Dependent Riccati Equation Contro o Formation Fying Sateites 1 Chang-Hee Won and Hyo-Sung Ahn Department o Eectrica Engineering University o North Dakota 15 Juy 4 Abstract Precise maneuvers o ormation ying sateites require a genera orbita dynamic equation and an eective noninear contro method. In this paper, noninear orbita dynamics o reative motion equations are derived or a constant distance separation ormation ying probem. This genera orbita dynamic equation aows eiptic, noncopanar, and arge separation distances between spacecrats as we as traditiona circuar, copanar, and sma separation distance cases. Furthermore, or the in-pane ormation ying scenario with arge constant ange o separation between sateites, we derived the change in position and veocity equations. A noninear contro method caed the state-dependent Riccati equation contro method is utiized to sove the ormation ying contro probem. This nove contro method or a noninear system aows the intuitive design tradeo between the contro action and the state error simiar to the 1 Chang-Hee Won and Hyo-Sung Ahn, Noninear Orbita Dynamic Equations and State- Dependent Riccati Equation Contro o Formation Fying Sateites, Journa o the Astronautica Sciences, Vo. 51, No. 4, October-December 3. 1

2 cassica inear-quadratic-reguator contro method. Two numerica simuations demonstrate the eectiveness o the new state-dependent Riccati equation contro method with the newy deveoped reative motion equations.

3 Introduction Mutipe spacecrat ormation ying is one o the key technoogies o current and uture space missions. The Nationa Aeronautica and Space Administration's Earth Observing-1 sateite demonstrated ormation ying technoogy by ying in a ormation one minute behind Landsat-7 in November, and Jet Propusion Laboratory is panning the Deep Space-3 mission to ormation y three separate spacecrats to take space optima intererometer measurements. The objective o ormation ying is to autonomousy contro two or more sateites reative to another sateite with minimum ground contro station invovement. There are two main chaenges in ormation ying technoogy. First, there has to be genera noninear reative motion equation and second, there has to be an appropriate noninear contro method to design an optima controer. So, in this paper, we derive a noninear orbita dynamic equation and we propose to use the State-Dependent Riccati Equation (SDRE) contro method to sove the ormation ying probem. On the orbita dynamic side o ormation ying, various studies have been reported. In 1985, Vassar and Sherwood studied ormation keeping or a pair o sateite in a circuar orbit in spite o disturbances such as aerodynamic drag and soar radiation pressure [1]. Kapia et a. deveoped and used inear Cohessey-Witshire (CW) equation or circuar orbit ormation ying in their paper []. They utiized inear puse contro method in their eampe. In, Yedavai and Sparks studied sateite ormation ying contro design based on hybrid contro system anaysis [3]. They derived a noninear version o CW 3

4 equation and inearized or sma reation disturbance between two sateites. But they assumed circuar orbits. Noninear CW equations or circuar orbits have been etensivey investigated by Ariend, Schaub, and Gim [4, 5]. Furthermore, Inahan, et a. derived a noninear reative motion equation and soved the probem by inearizing with sma curvature assumption [6]. The eect o noninearity and eccentricity are studied by Vaddi et a. [7]. Another popuar mode or ormation ying orbita dynamics is eader oowing mode o Wang and Hadaegh [8]. There, the eader sateite s rame becomes the ormation reerence rame. Then the states are measured and controed with respect to that reerence rame. Mesbahi and Hadaegh used eader oowing ormuations, graphs, inear matri inequaities, and switching theories [9]. However, the system is inear and with stateeedback contro. Our reative motion equations are noninear equations that mode eiptic, noncopanar orbita dynamics o ormation ying sateites. Various contro methods incuding bang-bang, inear pused contro, inear quadratic reguator (LQR), and hybrid contro have been studied or ormation ying [1,, 3]. However, a these methods were based on a inear contro theory with a inear system. In this paper, we utiize a noninear contro method known as the SDRE contro method. There are many noninear contro methods avaiabe in the iterature incuding eedback inearization, gain scheduing, and siding mode contro, however, they determine the contro aw by some anaysis method. Thus, it is diicut or a contro designer to intuitivey design a controer that is appropriate or the ormation ying. One nove noninear contro method caed State-Dependent Riccati Equation (SDRE) contro has the intuitive design advantages simiar to the cassica inear-quadratic-reguator (LQR) contro 4

5 method. The SDRE method is anaogous to cassica LQR method and the abiity to tradeo between contro eort and state errors is aso simiar. However, this method does not guarantee goba asymptotic stabiity and may give a suboptima controer. The controabiity and stabiity issues in SDRE contro are addressed by Hammett et a [11]. In 1998 Mracek and Coutier pubished a contro design or the noninear benchmark probem using the SDRE method [1], where they showed the eectiveness o SDRE method. We wi appy this nove SDRE method to ormation ying sateite appications and demonstrate its eectiveness through simuations. In the net section, we deveoped more genera ormation ying orbita dynamics or noncopanar eiptic orbits that is vaid or arge separation distance between ormation ying sateites. Then as a specia copanar case, we derive the equations necessary or constant ange separation ormation ying in the same orbita pane. Then we describe the SDRE method in more detai incuding the stabiity properties. Finay, we present simuation resuts o mutipe spacecrat ying in a ormation using SDRE and newy deveoped reative motion equations. Constant Separation Distance Orbita Dynamics Here we consider noncopanar and eiptic orbit geometry or constant separation distance ormation ying. Consider an eiptic orbit with the Earth center at the one o the oci as shown in Fig. 1. An orthogona coordinate rame, caed the s-rame, is attached to the eader spacecrat and moves with the î -ais which is radiay outward rom the center 5

6 o the earth. The ˆk -ais is out o the paper, and the ĵ -ais competes a right hand triad. The vector position o the oowing spacecrat is r r ρ = +. V Leader spacecrat ĵ ˆk î ρ Foower spacecrat r Mean motion o the eader (n) r True anomay o the eader ( ) ν Center o the Earth (Focus) Perigee Fig. 1. Eiptic Orbita Geometry. We wi assume that the inertia reerence rame, caed i-rame, is attached to the Earth center, and the eader spacecrat is in our eiptic orbit around the Earth with the μ mean motion n =, where 3 a 14 μ = m 3 /s and a is the semi-major ais o the orbit. The orbita dynamics o a spacecrat reative to the Earth is given as μ r + r = F, (1) 3 r where F represents the speciic eterna disturbance and/or speciic contro orces (orces per unit mass). I we et ρ = iˆ+ yj ˆ+ zkˆ, () 6

7 then the oower spacecrat dynamic is given as where μ ( ) r + ρ + r + ρ = F, (3) 3 r + ρ F denotes the speciic eterna disturbances and/or speciic contro orce (orce per unit mass). For an eiptic orbit, r r a(1 e ) /(1 + ecos ν ) = =. (4) The dynamic equation or the oower spacecrat with respect to the eader spacecrat can be written as ρ + μ ( r μ + ρ) r = F r + ρ 3 3 r F (5) Deine 3/ γ = [( r + ) + y + z ] (6) then 3 r ρ γ + =. Now we need to transer the acceeration ρ in the s-rame to the acceeration in the i- rame (inertia rame). The transormation is we known, or eampe see Wiese s book [14]. ρ = ρ + ω ρ + ω ρ + ω [ ω ρ] (7) i s s si s s si s s si s si s The second term on the right hand side is known as the Coriois acceeration and the ast term is known as centriuga acceeration. si ω is the anguar veocity o the eader sateite in the s-rame with respect to the i-rame, 7

8 si ω n(1 + ecos ν ) = 3/ [ (1 e ) T k ] T ν. This anguar veocity o the eader sateite varies with time. Consequenty, we have Aso we have Thereore we obtain si n(1+ ecos ν)( esin ν) T ω = ν 3/ [ νk ] (1 e ). T ρ = y, ρ = y, ρ = y. (8) z z z s s s νy νy ν ν y ν y ν = = + + ν y. (9) z z i ρ y ν ν ν y y ν ν Substitute Eq. (9) into Eq. (5), νy νy ν r r μ μ y ν ν ν y y = F F 3 γ, (1) r z z which simpiies to μ μ νy νy ν + ( r + ) γ r Fi μ y ν ν ν y y F = j γ. (11) Fk μ z+ z γ This is a genera noninear equation o spacecrat reative position dynamics. 8

9 F, F, and F are the resuting speciic disturbance or contro orce. We can rewrite Eq. i j k (11) in state-dependent coeicient orm. =, =, = y, = y, = z, = z, then we have Let or 1 μ 1 ν ν ν 1 γ 1 μ μr 1 Fi r γ 3 3 = μ + F j ν ν ν γ 5 5 Fk μ γ (1) A ( ) + BF ( ) + E ( ). (13) This equation is a genera noninear reative motion equation or ormation ying sateites. Note that Eq. (1) is not a unique equation. It is important to note that this noninear reative motion equation did not assume that r is much greater than the reative distances, y, and z. Aso this equation is vaid or eiptic and noncopanar orbits. However, to use SDRE method, we wi assume E( ), and in the simuations we wi check the vaidity o this assumption by epicity cacuating E( ). Furthermore, we can et target and target be the oower spacecrat s target position and speed in i -direction; y and y be the oower spacecrat s target position and target target speed in j -direction; and z and z be the oower spacecrat s target position and target target 9

10 speed in k -direction. The state errors are deined by 1 = 1 target, = target, 3 = 3 ytarget 4 4 targ notation,, = y et, 5 = 5 ztarget, and 6 = 6 z target. We wi use the [ ] =, (14) T target = target target ytarget y target ztarget z target, (15) d = A + A BF target +, and u = K. (16) dt Copanar Constant Ange Separation Orbita Dynamics The objective o ormation ying in this section is to achieve the constant ange between the eader and oower sateites in the same orbita pane. The reative position and reative veocity equations o the oower sateite with respect to the eader coordinate rame (s-rame) are derived. Fig. shows the ormation ying geometry with a constant separation ange, α, between the sateites. To keep this separation ange constant, the reative position and veocity are controed. 1

11 V V 1 î ĵ V 1 α θ Leader V β r ρ ν α V (, y ) r V y Foower ν V 1 θ V (, y ) 1 1 Perigee Fig.. Constant Ange Separation Formation Fying Geometry. In Fig., the distances rom the center o the Earth to sateites, r and r, are cacuated rom the true anomay. To keep the ange, α, constant, we contro the reative distance and veocities in the s-rame. The initia position coordinate, ( 1, y 1), o the oower sateite is the current position o the oower sateite with respect to the eader sateite s- rame. The target position coordinate, (, y ), o oower sateite is the target position o the oower sateite with respect to the moving eader sateite to keep the constant separation ange, α, in the same pane. Now, we derive the reative position reationship. The reative position o the oower sateite with respect to the eader sateite in s-rame is given as oows. ( ν) = ρ( ν) cos( β( ν)) iˆ, and y( ν) = ρ( ν) sin( β( ν)) ˆj (17) where, ρ( ν ) denotes the magnitude o the vector ρ( ν ), ρ( ν) r ( ν) r ( ν) r ( ν) r ( ν)cosα = +, 1 sinα βν ( ) = sin r ( ν) ρ ( υ ), (1 ) ( ) a r e ν = and 1 + ecosν (1 ) ( ) a r e ν =. (18) 1 + ecos( α ν ) 11

12 We derive the equation or the sateite s reative veocity in s-rame as s Δ V( ν ) = V ( ν) V ( ν ). (19) To ind the oower sateite s veocity vector in the s-rame, s V, we transorm the oower sateite s veocity vector to the s-rame. s V s V ( ν ) cosα sinα V ( ν ) ( ν ) = s = Vy ( ν ) sinα cosα Vy ( ν ) () where V ( ν) V ( ) T y ν is the veocity vector o oower sateite with respect to the s s eader sateite s s-rame. Fig. 3 shows this reationship graphicay. From Fig. 3, we know that V ( ν ) = V ( ν)cosθ and V ( ν ) = V ( ν)sinθ where V ( ν ) is the magnitude o the y s sateite s veocity vector to the s-rame, V ( ν) = V ( ν) = V ( ν) + Vy ( ν). Then substituting these equations to Eq. (), we obtain s V s V ( ν ) V ( ν )cos( α θ) ( ν ) = s = Vy ( ν ). (1) V ( ν )sin( α θ) V 1 î s V y α θ s V V 1 ĵ Fig. 3. Transormation o the Foower Sateite Veocity Vector to the s-frame. 1

13 1 Vy We can determine θ by θ = tan where V V μ μ ( ν ) = ( esin ν ), and V y ( ν ) = (1+ ecos ν ) a(1 e ) a(1 e ) () where ν = ν α. Net, we derive the equation or the eader sateite veocity vector, V ( ν ) = ( e sinν ) a μ e (1 ) V, iˆ μ, and V ( ) (1 cos ) ˆ y ν = + e ν j. (3) a(1 e ) So the reative veocity o oower sateite with respect to the eader sateite is s V( ν) ( V ( ν) V( ν) )ˆ Δ ( ν) = ( ν) ( ν) j. (4) s Δ = i, and V ( V V ) ˆ y y y These position and veocity vectors o Eq. (17) and Eq. (4) are controed to keep a constant separation ange between the sateites. I we assume that the true anomay, ν, varies in discrete steps, then using Eqs. (17), (1), and (4) the target states are deined as ( ν k+ 1) = ρν ( k+ 1) cos( βν ( k+ 1)), y( ν k+ 1) = ρν ( k+ 1) sin( βν ( k+ 1)), Δ V ( ν ) = V ( ν )cos( α θ ) V ( ν ) +1, k+ 1 k+ 1 k+ 1 k Δ V ( ν ) = V ( ν )sin( α θ ) V ( ν ) +1, (5) y k+ 1 k+ 1 k+ 1 y k and the initia states as 1 ( ν k) = ρν ( k) cos( βν ( k)), y1 ( ν k) = ρν ( k) sin( βν ( k)), Δ V1 ( ν k) = V1 ( νk)cos( α θk) V1( νk), and V1y ν k V1 νk α θk V1y νk Δ ( ) = ( )sin( ) ( ). (6) 13

14 The position and veocity change necessary to keep a constant separation ange are given as oows. Δ = ( ν ) 1( ν ), y y ν y1 Δ = ( ) ( ν ), Δ V =ΔV ΔV1, and Vy Vy V1 y Δ =Δ Δ. (7) Here we are inding the dierence between the net target position and veocity o the oower sateite and the current position and veocity o the oower sateite. We wi try to ind the optima controer that minimizes this error. Finay, the net state errors in Eq. (14) are epressed as oows ( i+ 1) = ( i) +Δ( i), (8) where = [ ] T and Δ= Δ ΔV Δy ΔVy Δz ΔVz. State-Dependent Riccati Equation Noninear Contro Theory The SDRE method started in 1996 by James Coutier and his coworkers [1]. The SDRE contro method reormuates a noninear dynamic equation into state-dependent coeicient structure and soves a SDRE as in cassic inear quadratic reguator optima contro. It is intuitivey simiar to the LQR method in terms o the tradeos between the contro eort and the errors. Aso, the SDRE method has good robustness properties as the LQR method. In this paper, to contro the noninear system represented by the noninear orbita dynamic reative motion equation, we utiize the SDRE method. The SDRE concepts wi be summarized in this section, and or more detaied description reer to Coutier et a [1]. Consider a genera noninear dynamic system. 14

15 = ( ) + g( ) u, and y = C( ) (9) where n m Ñ, u Ñ. We assume that () = and g ( ) or a. Remark: From Eq. (1) we note that or our probem () = is not generay satisied, however, assuming ρ is sma we can assume () =. The optimization probems is to ind the contro, u, that minimize the cost unction, 1 T T J = Q( ) + u R( ) u dt (3) subject to the noninear dierentia equation, (9), where ( ) k k Q C, R( ) C, Q ( ), R( ) > or a. The SDRE method obtains a suboptima soution o the above probem. First, put the noninear Eq. (9) to the state-dependent coeicient orm, = A ( ) + Bu ( ), (31) using direct parameterization, where ( ) = A( ) and g( ) = B( ). We note here that the choice o A ( ) is not unique, and this may ead to a suboptima controer. Aso this does not impy variations o the A ( ) matri as a unction o, but we are simpy putting ( ) into a orm that ooks ike a inear orm. Second, sove the SDRE state-dependent Riccati equation Q A P PA PBR B P T 1 T T ( ) + ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ) = (3) to obtain P ( ). Third, obtain the noninear controer o the orm. 1 T u ( ) R ( B ) ( P ) ( ) = (33) We note that the Riccati matri, P ( ) depends on the choice o A ( ), and because A ( ) is not unique, we have mutipe suboptima soutions [1, 13]. 15

16 In addition Coutier proved that i A( ), B( ), Q( ), and R( ) are smooth, and the pair[ ( ), ( )] is pointwise stabiizabe, and the pair A B [ ( ), ( )] C A is detectabe in the inear sense or a, then the SDRE method produces a cosed oop soution which is ocay asymptoticay stabe [1,1]. We can check stabiizabiity and detectabiity by orming the controabiity and observabiity matrices and checking their rank. Orbita Mechanics Preiminaries Beore we present the simuation resuts, we review some o the orbita mechanics concepts that are necessary to do the simuations. A. Position o the Sateite as a Function o Time We wi ind the position o the spacecrat as a unction o time, r ( ) t. Assuming at t = the spacecrat is at perigee, we determine eccentric anomay, E, using the Keper s equation. M = E esin E = nt (34) where M is the mean anomay, n is the mean motion, t is time, and e is the eccentricity. In order to do this, we ind the initia eccentric anomay by E M (1 sin u) + usin M = 1+ sinm sinu, (35) where u = M + e. We aso have the oowing equations. F( E ) = E esine M, and i i i df( Ei ) = ecos Ei, de i 16

17 Then using Newton s method iterativey, we ind E (ina E is denoted by E ) by i E i+ 1 FE ( i ) = Ei, and we determine df / de i df( E) r () t = a = a(1 ecos E). (36) de B. Orbit Perturbation Modes The obateness (J), soar radiation pressure, and air drag orce are considered as the eterna disturbance. We wi use the oowing simpe modes. These disturbances are added to the contro orce because eterna disturbances. Obateness (J) F in Eq. (1) represents the speciic contro orce and The perturbing acceerations due to J may be given in the ijk -coordinate system as [15] oi μjr = 1 ( sin i) (1 cos u) 3 E 3 4 r (37) 3 μjr oj = ( sin i) (sin u) r, (38) E 4 3 μjr =, (39) r E [ sin icosisin u] ok 4 where argument o atitude is u = ν + ω (true anomay and argument o perigee), is equa to.1863, i is orbita incination, μ is gravitationa constant, R E is the equatoria radius, and r is the distance rom the center o the Earth to the eader sateite. J 17

18 Atmospheric Drag The aerodynamic orce per unit mass due to drag on a sateite is given by S ρ C V m d =.5 d, (4) where ρ is atmospheric density, C d is drag coeicient, S is sateite cross sectiona area, m is mass, and V is sateite s veocity [16]. For the eader sateite in eiptic orbit, the veocity is given as V = ν r. And the oower sateite veocity is cacuated by 4 6 V = ν r (41) where r is given in Eq. (4) and, 4, and 6 are given in Eq. (1). This drag orce wi aect the veocity in j direction ony. Soar Radiation Pressure [14], A simpe mode o soar radiation pressure (orce per unit mass) is given as oows s S r m 6 = (1 + ), (4) where r is a reection actor (the reection actor is assumed 1 or the worst case), S is sateite cross sectiona area eposed to the Sun in m, and m is sateite mass in kg. We wi assume the mode or summer sostice where the sun ange, θ, is 3.5 degrees. We wi aso assume that the sateite is at the perigee. The disturbance orces due to the soar radiation pressure on each ais o the s-rame are determined by using the inormation o orbit incination ange and true anomay ange as shown in Fig. 4. It is assumed that 18

19 initiay, the negative i -ais and the soar radiation incident direction ater projected on the ij -pane are the same. Then as the sateite moves the i -ais moves with the true anomay, ν, as shown in the top circe o Fig. 4. ĵ ĵ ν î ij î ˆk î s θ Fig. 4. Soar radiation Pressure Geometry. From Fig. 4, we determine the soar radiation pressure in ij-rame as, and the soar radiation pressure in each o the s-rame is given as = cos( θ ) (43) ij s = cos( ν ), = sin( ν), and = sin( θ ) (44) i ij j ij k s where s is the soar radiation pressure rom Eq. (37). Formation Fying Simuations In order to appy the newy deveoped ormation ying orbita dynamic equations and 19

20 SDRE noninear contro methods, we perorm three dierent computer simuations. The irst simuation is or sateites in Moynia orbit and 5 meters separation in a three ijk - directions. The second simuation is or a arge separation distance ormation ying with constant ange o separation equa to 1 degrees. A. Moynia Orbit Formation Fying In this simuation, two sateites are in Moynia orbit: semi-major ais o 3,978 km, eccentricity o.75, and incination o 63.4 degrees. Thus this is ormation ying in noncircuar and noncopanar orbit. Because we assume ony 5-m separation distance, μ r μr γ is very sma, in the order o 6 1 to 1 7, and the assumption E( ) is satisied in Eq. (1). Furthermore, we check or the controabiity o the system by check the rank o [ A ( ), B ( )]. Figure 8 shows the Moynia orbit with respect to the Earth.

21 1 9 Distance (km) Sateite Orbit 3 Earth True Anomay (deg) Fig. 8. Formation Fying Sateite Orbit with respect to the Earth. Initiay, both sateites wi begin at the same position ( =, y =, and z = ), and then they wi orm the ormation o 5 meters apart in a three directions o the s-rame (i.e, = 5, y = 5, z = 5 ). We assumed summer sostice with the sun ange o 86.5 degrees. We used soar radiation pressure, air drag and J perturbations. The simuation parameters are summarized in Tabe 1. The thrusters are assumed to be 1N thrusters in ijk directions. I the thrust orce required to keep the ormation is greater than 1N then ony 1N is appied in the required direction. 1

22 Tabe 1. Sateite Parameters or Formation Fying in Moynia Orbit. Leader Sateite Foower Sateite Mass 155 kg 41 kg Dimension m m 3 Air drag coeicient.. Soar reectivity coeicient (r).6.6 We chose the integration time step to be.5 second and simuation duration was 7 hours. We used the contro weighting matri, R diag[ 1 1 1] matri, Q diag[ ] =. =, and the state weighting i-direction (m) j-direction (m) k-direction (m) Time (Hours) Fig. 9. Separation Distances o Two Formation Fying Sateites.

23 Figure 9 shows the separation distances decreases graduay to 5 m in ijk - directions. We note that in the i -direction the sateite reaches the target position o 5 meters irst in about 3 hours, the k-direction second in about 4.5 hours, and the j-direction ast in about 6 hours. The required time to reach 5 meters is arge because we are imiting out contro orce to be 1N. Figure 1 shows the corresponding thrust puses. We note that maimum thrust eve o 1N was ired unti the sateite reaches the desired ormation. i-direction (s) j-direction (s) k-direction (s) Time (Hours) Fig. 1. Thrust Puses o the Moynia Orbit Foower Sateite. 3

24 B. Constant Ange Separation Formation Fying in the Same Pane In this simuation, two sateites are in ow earth sun synchronous orbit: atitude o 8km, eccentricity o.4, and incination o 98.6 degrees. These two sateites wi keep a constant separation ange o 1 degrees with each other. We assumed summer sostice with the sun ange o 1.1 degrees. We used soar radiation pressure, air drag perturbations and J perturbations. In this case the idea thruster was assumed. We chose the integration time step to be 1 seconds and simuation duration was 1 hour. We used the contro weighting matri, R = I 33, and the state weighting matri, Q = I 6 6. The system is checked or the controabiity in the simuation using Matab rank μ μr and ctrb unctions. In this simuation, the separation distance are arge so r γ and the assumption E( ) is not satisied in Eq. (1). The error is in the order o 1 to However, we wi assume E( ) and use the SDRE method to contro the ormation. Thus we are inding the optima controer or an approimate probem o A ( ) + BF ( ). 4

25 (km) y (km) z (km) Time (Seconds) Fig. 11. Separation Distances o Constant Ange Formation Fying Sateites. Figure 11 shows the reative distance between the eader and the oower sateites when the ange between them is kept at a constant. Note that the sateites move in a sinusoida ashion in i -direction and j -direction. And the distance in k -direction is varying more rapidy. We are assuming 1N thrusters in a three directions. Because the thrust necessary to keep the ormation is arger than 1N, the ange cannot be kept at 1 degrees or too ong. The thrust necessary or this ormation ying is shown in Figure 1. Note that we require a arge amount o thrust to achieve this ormation ying. Finay, Fig. 13 shows the separation ange between the two sateites. We are trying to keep this ange to 1 degrees in this ormation ying simuation. Note that the separation ange is kept 5

26 with ±.1 degrees or irst 36 seconds. i-direction (m/s ) j-direction (m/s ) k-direction (m/s ) Time (Seconds) Fig. 1. Thrusts o the Constant Ange Formation Fying Foower Sateite. 6

27 Separation Ange (Degrees) Time (Seconds) Fig. 13. Separation Ange Variation. Concusions Noninear orbita dynamic equations or ormation ying sateites are deveoped. These reative motion equations are vaid or noncopanar and eiptic ormation ying. For the specia case o in-pane ormation ying with arge ange o separations, we derived the change in position and veocity equations that can be used or ormation ying. In order to contro the sateites modeed by the newy deveoped noninear equations, we utiized the State-Dependent Riccati Equation (SDRE) contro method. Computer simuations veriied that the ormation ying using the newy deveoped orbita dynamic equations and SDRE method gave avorabe resuts. 7

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29 [11] Key D. Hammett, Christopher D. Ha, and D. Brett Ridgey, Controabiity Issues in Noninear State-Dependent Riccati Equation Contro, Journa o Guidance, Contro, and Dynamics, Vo. 1, No. 5, September-October 1998, pp [1] Curtis P. Mracek, and James R. Coutier, Contro Designs or the Noninear Benchmark Probem Via the State-Dependent Riccati Equation Method, Internationa Journa o Robust and Noninear Contro 8, 1998, pp [13] Evrin Erdem and Andrew Aeyne, Design o a Cass o Noninear Controer via State Space Dependent Riccati Equations, IEEE Transactions on Contro Systems Technoogy, Vo. 1, No. 1, January 4, pp [14] Wiiam E. Wiese, Spaceight Dynamics, McGraw-Hi Pubishing Company, [15] Vadimir A. Chobotov, editor, Orbita Mechanics, Second Edition, AIAA, Virginia, 1996, p.8. [16] James R. Wertz and Wiey J. Larson, Space Mission Anaysis and Design, Third Edition, Microcosm Press, Kuwer Academic Pubishers,