Dynamic Model of the Spacecraft Position and Attitude

Transcription

1 Dnami Model of the Spaeraft Position and Attitude Basilio BNA, nrio CANUT Dipartimento di Automatia e Informatia, Politenio di Torino Corso Dua degli Abruzzi 4 9 Torino, Ital tel , fa bona@polito.it Referene Frames. Referene Frames Desription In this report the following referene frames will be onsidered:.. Inertial Referene Frame IRF (,,, ) R i k is an arth entered equatorial inertial frame, with: at the entre of the arth. i along the intersetion of the mean elipti plane with the mean equatorial plane, at the date of //; positive diretion is towards the vernal equino. k orthogonal to the mean equatorial plane, at the date of //; positive diretion is towards the north. ompletes the referene frame... arth-fied Referene Frame FRF (,,, ) R i k is an arth fied non-inertial frame, and shall be the most reentl defined International Terrestrial Referene Frame (ITRF). We assume: at the entre of the arth. i along the intersetion of the equatorial plane and the Greenwih meridian plane; positive diretion from the earth entre to the longitude point on the equator. k orthogonal to the equatorial plane; positive diretion towards the north pole. ompletes the referene frame...3 rbit Plane Referene Frame PRF (,,, ) R i k is a loal non-inertial orbit plane referene frame, with: P P P P P at the entre of the arth. P i along the intersetion of the orbit plane with the equatorial plane. P k obtained b rotating k positivel around i P P b the orbit inlination angle i. Dnamis_revised.do data reazione /7/ Pagina di 8

2 ompletes the referene frame. P..4 Loal rbital Referene Frame LRF (,,, ) R i k is a non-inertial loal orbital referene frame, with: at the atual satellite entre of mass. i (roll) parallel to the instantaneous diretion of the orbital veloit vetor = diretion towards positive veloit; i = v v. v r; positive (pith) parallel to the instantaneous diretion of the orbital angular momentum h = r v, where r v r is the vetor from the arth entre to ; = r v. v h k (aw) ompletes the referene frame: k = = i. v h..5 Spaeraft Referene Frame SCRF (,,, ) R i k is a loal satellite non-inertial frame, with: SC SC SC SC SC at the entre of the launher-satellite adaptor interfae plane SC i along the launh vehile ais; positive diretion is towards the launh vehile nose. SC k orthogonal to the satellite earth fae; positive diretion is towards nadir. SC ompletes the referene frame. SC..6 Attitude Control RF RF (,,, ) R i k is a loal non-inertial satellite referene frame, with: at the atual satellite entre of mass (CM). i,, k parallel to i SC, SC, k SC.. Transformation Matries between Referene Frames In the following we use the notations (, α), (, β), (, γ) R i R R k to represent, respetivel, rotations around the X, Y, and Z ais. We remember that a rotation about fied aes pre-multiplies, while a rotation about the moving aes post-multipl. a) R R, given b matri P R : P s s s Ω Ω i Ω i R = R P ( k, Ω ) R( i, i) = s s Ω Ω i Ω i (.) s i i Dnamis_revised.do data reazione /7/ Pagina di 8

3 notie that, while the inlination angle i is onstant (see Table ), the right asension Ω varies at a -7 rate of.9856 deg/da ( rad/s), to allow the satellite to be sun-snronous R b) R, given b matri R : s δ δ R = R ( k, δ) = s δ δ (.) notie that the angle δ varies at a rate of 36 deg/da ( rad/s), due to dail rotation. R ) R, given b matri R R : where v and r are defined in Setion..4. d) R R, given b matri P v r v v r v = v r v v r v R : P (.3) R = R R = P P ( R) R = P s s ss ss s s ss ss δ δ Ω Ω i Ω i + + δ Ω δ Ω δ Ω i δ Ω i δ Ω i δ Ω i s s s s s ss sss s δ δ Ω Ω i Ω i = + + δ Ω δ Ω δ Ω i δ Ω i δ Ω i δ Ω i s s i i i i e) R R, given b matri P where α is the true anomal. f) R R, given b matri P R : (.4) s α α P R = R ( i,9 ) R(,9 + α ) = s α α (.5) R : ss s s ss + ψ ψ ϕ ψ ϕ ψ ϕ ψ ϕ R = R ( k, ψ) R(, ) R( i, ϕ) = s s s s s s s + ψ ψ ϕ ψ ϕ ψ ϕ ψ ϕ (.6) s s ϕ ϕ where ϕ = roll angle, = pith angle, ψ = aw angle; for small angles the above matri redues to: ψ R = ψ ϕ (.7) ϕ Dnamis_revised.do data reazione /7/ Pagina 3 di 8

4 Dnami State Spae Model. Assumptions The state spae model derived in this report desribes a) the dnamis of the S/C CM position with respet to the inertial frame R (orbital dnamis); b) the dnamis of the S/C referene frame R with respet to the orbital frame R (attitude dnamis). The following assumptions hold: Assumption : the dnamis of the CM position is desribed b the differential equations of the origin represented in R b the vetor r = z. The CM veloit origin is represented in R b the vetor v = v v v z. Assumption : the attitude of α = ϕ ψ impliitel defined in eqn. (.6). R R with respet to R is given b the three roll-pith-aw angles Assumption 3: in order to transform the vetors represented in a generi referene frame represented in another referene frame R, the following relation holds: B RB RA R into vetors A B p = R A p (.8) where p represents a generi vetor in the referene frame B R and R is a orthonormal matri with R B A B positive unitar determinant (proper rotation). Assumption 4: inputs and/or measurements, defined or obtained in referene frames other than the referene frame R, are assumed to be alread represented in this referene frame. Assumption 5: the aes i,, k of the A/C referene frame are assumed to oinide with the prinipal aes of inertia. Assumption 6: the S/C mass m and inertia matri = (.9) z are assumed to be time-invariant, although this is not true, due to propellant onsumption.. Plant Dnami quations In order to model the S/C dnamis it is neessar to introdue first the orbital kinemati equations... Kinemati equations Assumption 7. The total inertial veloit vetor ω is the sum of the orbital rotation veloit and the S/C attitude rate. Dnamis_revised.do data reazione /7/ Pagina 4 di 8

5 To this end let us denote R i k the loal orbital frame,. with (,,, ). with (,,, ) R i k the same frame rotated around the z-ais b the angle ψ, 3. with (,,, ) R i k the previous frame rotated around the -ais b the angle, R i k the attitude frame, obtained b rotating the previous frame around 4. with (,,, ) the -ais b the angle ϕ, 5. with ω Then we an write = ω = r v the orbital rotation veloit. r ω = ω + ψk + + ϕ i Upon transformation of the vetor omponents in the attitude referene frame, it results ω ϕ ω = ω (, ϕ) (, ) (, ψ ) ω (, ϕ ) = R i R R k + R i ω ψ z and epliitl ω s s ϕ ψ ω = sss + ω + s ψ + ψ ϕ ψ ϕ ϕ ϕ (.) ω ss s s z ψ ϕ ψ ϕ ϕ ϕ The kinematis non linear equations are therefore ϕ = s ψ + ω s ω ψ + s = ω sss + ω sψ sω + sψ + s ss s ω ϕ ϕ ϕ ϕ ψ ϕ ψ ϕ ϕ ϕ ϕ z ϕ ϕ ϕ ψ ϕ ψ ϕ ( ) ( ) = ω s ω ω ϕ ϕ z ψ ψ = ω + s ω s ω ω s s s ω ( ) ( ) ( s s s ϕ z ϕ ψ ) ϕ z ϕ ϕ ϕ z ψ ψ ϕ ψ ϕ ψ = ω + ω ω s ϕ ω ω ω ω ω = ( + s s s z ) + s ϕ ϕ ψ ψ ( s ϕ ϕ ) The final form of non linear equations is the following s s = ω + ω + ω ω s s ϕ = ω + ω + ω ω ( s ϕ ϕ ) ψ z = ω s ω ω ϕ ϕ z ψ ψ = ω + ω ω ( s s s ϕ z ϕ ψ ) ψ z (.) (.) Dnamis_revised.do data reazione /7/ Pagina 5 di 8

6 Beause attitude will be ontrolled, we will adopt a quasi linear form, b negleting terms of the third order and more: ϕ = ω + ω ψω z = ω ϕω ω z ψ = ω + ϕω ψω z (.3).. rbital Dnamis The differential equations desribing the S/C CM dnamis are given b the Newton equation: F t + mg t ma t = (.4) total where F = total F is the vetor sum of all disturbane and ommand fores, eept those due to gravit k k and inertia, ating on the S/C CM, g is the loal gravit aeleration vetor ating on the S/C CM, a = v and v are the S/C CM linear aeleration and veloit respetivel, and m is the S/C mass. We note that aeleration ( t) a an be deomposed into two terms: a) Centripetal aeleration a t, due to the S/C orbital angular veloit T 537 s. This aeleration is given b a = ω,nominal ( ω ) entrifugal fore F ( t) =ma. π ω =, with T r and gives origin to a The vetor r =r ρ gives the S/C position relative to the Inertial Referene Frame R, where ρ is the radial versor and r nominal = R + h S 668km; ω has been previousl defined. For a irular orbit ρ =k. b) Residual aeleration t = t t res a a a. We further assume that the eternal fores ating on the S/C onsist onl in:. Drag fores F ( t ), due to aerodnami and magneti drag; we assume to possess the drag fores d omputed from GMV simulations and represented in R ; it is threfore neessar to represent them in the inertial referene frame, as follows: F t F t d d, GMV t = F t F t d d = dgmv, F t F t dz dz, GMV R R F R (.5) where R is given in eqn. (.3).. Command fores due to atuation of nine thrusters =,,, 8 whose thrusts are olleted in a vetor ( t) u : t Dnamis_revised.do data reazione /7/ Pagina 6 di 8

7 u t t F t, u t t, t t t F t z, u t t 8 F t = F t = Vu t = V (.6) where V is the ation matri to be defined later. t Assumption 8: we assume that the ommand fores are represented in the attitude ontrol referene frame R, so it is further neessar to transform them into inertial frame as follows where F t F t = F t = R F t (.7) F t z R = R R (.8) The gravit vetor g t, due to zonal harmonis indued b arth oblateness, an be approimatel written as: μ 3 R μ g( r, δ) 3 (( 5sin δ ) r sinδ ) 3 res r r = + r k r g (.9) r r where k is the earth pole ais. Finall, eqn. (.4) an be rewritten as: μ d 3 res res t t m m t m t m t F + F r+ g a a = (.) r Assuming a nominal orbital motion with angular veloit where fore: ω, realling the triple produt rule a b = ( a ) b( a b) (.) a b denotes the salar produt a b = a b, and noting that ω r, it follows that the entripetal ( ) μ F = m a = m ω ω r = m ω r = m r (.) r Ο 3 where 3 π r ω = and T = π, we have a nominal equilibrium between F and the first right-side Ο S μ T S term of (.9) Then, eqns. (.4) and (.) an be rewritten as Dnamis_revised.do data reazione /7/ Pagina 7 di 8

8 so that F t + F t ma t + mg t = d res res,,,, F t F t a t g t d res res F t F t m a t m g t d + res + res = F t F t a t g t dz z res, z res, z a = F F F F g m a = F F F F g m a = F + F + F + F + g m res, d d dz res, res, d d dz res, res, z d d dz z res, z (.3) (.4)..3 Attitude Dnamis The differential equations desribing the S/C attitude dnamis are given b the uler equation: where t k ( t k k t ) a t ω t ω t k C + b F a = (.5) k C is the vetor sum of all disturbane and ommand torques, eept those due to gravit and dk k inertia, ating on the S/C CM, b is the vetor of appliation of F, is the inertia matri, ω is the S/C k dk inertial angular veloit, and a a is the S/C referene frame angular aeleration. Contrar to the orbital dnamis equation, here all vetors are represented in R (see also Setion..). The following assumptions hold: Assumption 9: the onl torques ating on the S/C are due to aerodnami drag, magneti drag and thrusters atuation. Gravit gradient torques are assumed negligible. Assumption : gravitational fores, entrifugal fores F t and drag fores F d ( t ) have appliation points oiniding with the S/C CM, so the do not give origin to torques. With these assumptions, eqn. (.5) redues to where C is the ommand torque vetor due to thruster atuation ( t) Assumption 8): where the matri d a C + Ca ω ω = (.6) C V will be defined later. t As with the drag fores, the drag torques now it is neessar to trasform the as follows: t t u (alread represented in R, as in u t t t ut = = t t = t C z ut 8 t C t C t V u t V (.7) C are obtained from GMV simulation, and represented in d R, so Dnamis_revised.do data reazione /7/ Pagina 8 di 8

9 C t C t s s C t d d,gmv ψ ψ d C t ( ) C t ss s sss s C t d = R d,gmv = + ψ ϕ ψ ϕ ψ ϕ ψ ϕ ϕ d (.8) C t C t s+ss ss s C t dz dz,gmv ψ ϕ ψ ϕ ψ ϕ ψ ϕ ϕ dz R R R Then, eqn. (.6) an be rewritten as: Sine it follows: ω ω d a = + a C C (.9) ( ) ( ) ( ) ω ω ω ωω z z z ω ω = ω ω ω ω ω, z = z z (.3) ω ω ω z z ωω ( ) ( ) ( ) a s s C C ωω a ψ ψ d z z a ss s sss s C C a = ωω ψ ϕ ψ ϕ ψ ϕ ψ ϕ ϕ d z z a s+ss ss s C C az ψ ϕ ψ ϕ ψ ϕ ψ ϕ ϕ dz z ωω..4 State Spae quations (.3) The omponents of the sstem state are assumed to be the 6 linear positions and veloities of S/C CM i and the 6 angles and angular veloities around the three i,, k aes: = = z = 3 = 4 = 5 = z 6 = ; (.3) ϕ 7 = = 8 = ψ 9 ω = ω = = ω z From (.3), (.4), (.3) and (.3) we an write the differential equations in state variables form. Time dependene is omitted for brevit. Dnamis_revised.do data reazione /7/ Pagina 9 di 8

10 = 4 = 5 = 3 6 = F F F F g m = F F F F g m = F F F F g m = + ω 4 d d dz res, 5 d d dz res, 6 d d dz z res, z = ω 8 7 = + ω = 8 C + 9 d 8 s C 9 d s C 8 dz ( ) C + + z = ( ss s ) C ( sss ) C s C 7 8 ( ) C d d dz z = ( s +ss ) C + ( ss s ) C + C + d d 7 8 dz ( ) + C (.33) z z where = os ( t ), s = i i i sin( i t ). 3 Thruster ommand 3. Line of ations Let us denote with v the unit vetor of the mean line of ation of a thruster, defined as the mean thrust t diretion. It is defined b the following data in ase of the ion thruster (ITA): α π/+β β v t α Dnamis_revised.do data reazione /7/ Pagina di 8 v t β

11 Figure Angular oordinates of the mean lines of ations of ITA and MTA v α os α os β = sin α os β sin β =±.49 +Δα t β Δα Δβ = +Δβ.9 (3 σ).9 (3 σ) and b the following data in ase of miro thrusters (MTA): sin β os α v = os β os α t sin α α = π α +Δ α = π.35 +Δ α, =, 4 α = π Δ α, =, 3 α α =.35 +Δ α, = 5, 8 =.35 +Δ α, = 6,7 α =.35, β = +Δβ Δα Δβ.5 (3 σ).5 (3 σ) (.34) (.35) Remarks. Note that the diretion unertaint in ase of miro thrusters has not et been frozen. In the design ase, the ation matri V assumes the following form: t Dnamis_revised.do data reazione /7/ Pagina di 8

12 V = s t ± s s s s s s s s s = sin α = os α s = sin α = os α (.36) 3. Appliation points Denote with a the appliation point vetor of thruster. In ase of ion thruster it holds a a a. a = +Δ = +Δ a = a b a.tan α a = +Δ = +Δ a a =Δ z z Δa δ Δa δ Δa δ and in ase of miro thrusters it holds a z a a a, a b a, = +Δ = +Δ = a e a = ± +Δ a =± d +Δa z z Δa.3 (3 σ) Δa.3 (3 σ) Δa.3 (3 σ) z a =.76m b =.8m =.55m e =.8m (.37) (.38) In the design ase, the appliation matri V assumes the following form: t a a a b b a a b b V = b e e e e e e e e t (.39) d d d d d d d d 3.3 Command matri The ommand matri holds B relating the thrust vetor t u to fore/torque vetor F in spaeraft oordinates t Dnamis_revised.do data reazione /7/ Pagina di 8

13 In the design ase, it holds B t V v v v = = F F u t t t t8 Vt a v a v a v t t 8 t 8 t F u z t t t t C F = = Bu = B C u t8 C z (.4) B t s ± s s s s s s s s = d es d es d es d es d es d es d es d es (.4) as as bs bs as as bs bs a a b b a a b b 4 rbital Parameters Let now define the Kepler orbital parameters in order to be able of derive from them the si position and veloit omponents and vie-versa. 4. Kepler rbital Parameters Si parameters define an ellptial orbit aording to the Kepler laws:. the semi-maor ais a of the orbit ellipse; b. the orbit eentriit ε =, where b is the semi-minor ais; a 3. the inlination of the orbital plane with respet to the equatorial plane i; 4. the angle of right asension of the asending node (also known as RAAN) Ω, i.e. the angle on the equatorial plane from the vernal equino to the asending node. The asending node lies on the intersetion between the equatorial plane and the orbital plane, in orrespondene to the passage of the orbit from southern to northern emisphere; 5. the argument of the perigee ω, i.e. the angle from the asending node to the perigee, measured on the orbital plane: 6. the epoh of passage from the perigee t, i.e. a referene starting time. Fig. depits some of these parameters. Dnamis_revised.do data reazione /7/ Pagina 3 di 8

14 Fig. : the orbital parameters. It is helpful to define now an auiliar orbital plane referene R sstems wih differs from R sine its P P ais is aligned with the perigee and varies aording to its rotation: R is defined as follows P ( ) R, pqw,, is a loal non-inertial orbit plane referene frame, with: P P at the entre of the arth. P p points toward the perigee of the orbit plane. q is perpendiular to i and orresponds to a true anomal angle ν (see Fig. ) of P 9. w ompletes the referene frame. The transformation R R is given b the following matri P R = R( k, Ω P ) R( i, i) R( k, ω) = R R P ( k, ω) = ss s s ss Ω ω Ω i ω ω Ω Ω i ω Ω i = s s ss+ s p q w = + Ω ω Ω i ω Ω ω Ω i ω Ω i ss s i ω i ω i (.4) 4.. From Kepler Parameters to Position and Veloit The position r an be omputed from the Kepler parameters in the following wa: ( ε) ε r = a os p+ a sin q (.43) where is the eentri anomal angle (see Fig. ), obes the differential equation whih has a solution ( ε os) = 3 GM (.44) a Dnamis_revised.do data reazione /7/ Pagina 4 di 8

15 GM t ε sin t = M + 3 ( t t) (.45) a Fig. : the orbital plane and its parameters where M is the mean anomal at an arbitrar instant of time t. Notie that the mean anomal M hanges b 36 during one revolution, but, in ontrast to the true and eentri anomalies, inreases uniforml with time. The veloit v = r an be omputed as: where r = r. ( sin ε os ) GM a v = p+ q (.46) r 4.. From Position and Veloit to Kepler Parameters We assume to know the omponents of the versor w from (.4): w w = w w z The first two keplerian parameters i and Ω are omputed as: ( z) i = atan w + w, w ; ( w w ) Ω= atan, (.47) Dnamis_revised.do data reazione /7/ Pagina 5 di 8

16 where atan, tan = ( ) = = 9 if ; 9 8 if ; 8 9 if ; 9 if ; (.48) Net the semi-maor ais a is omputed as: a v = r GM (.49) where v = v. The eentriit follows: where the semi-latus retum p is given b: p ε = (.5) a and h = r v. The eentri anomal is now: where we have indiated The mean anomal therefore is h p = (.5) GM ( ) = atan r v a n, r a (.5) GM n = (.53) 3 a ε sin M t = t t (.54) with t the epohs of r and v. In order to find the remaining orbital parameter ω it is neessar to ompute the argument of latitude u; this is the angle between r and the line of nodes, i.e. u ω ν = + : and Finall ( ) u = atan r, rw + rw (.55) z ( ) ν = atan ε sin, os ε (.56) ω = u ν (.57) Dnamis_revised.do data reazione /7/ Pagina 6 di 8

17 4..3 Latitude and Longitude Let r be the vetor representing the S/C position with respet to R, i.e. the origin of R represented in R. For nominal motion r = R = R + h. After some omputation, the following relation holds: S S r ( ss ) ( ss ss) s + δ Ω δ Ω α δ Ω i δ Ω i α r = r ( s s ) ( ss ) s R = + + δ Ω δ Ω α δ Ω i δ Ω i α (.58) S r ss z i α R from whih it is possible to ompute the latitude and longitude of the S/C loal vertial, as follows: rz latitude = arsin in degrees r (.59) longitude = atan, in degrees ( r r ) 4..4 Small entriit rbits For an orbit with small eentriit ε, like the present one, the argument of the perigee ω is not well defined, sine small hanges in the orbit an ause large hanges in the perigee loation. It follows that the two referene frame vetors p and q in (.4) are not well defined as the an var signifiantl. Similarl the anomal (true or eentri) an var onsiderabl and even be non-monotoniall inreasing (with little phsial sense): in (.5) an var from π / to π / sine ra. In this ase it is better to assume as -ais vetor for anomal measurement the eentriit vetor e, defined as μ v h = r+ e (.6) r Realling that h = r v, the eentriit vetor e is well defined, sine, realling the triple produt in (.), results μ e = vr( v r) v r (.6) r ne e is normalized, it is possible to determine the -ais as w e e 4. Starting Conditions for the rbital Simulation Aording to GMV and Alenia douments, it is neessar to assume the perigee of the orbit on the equatorial plane (i.e. on or near the asending node) sine in this situation the worst ase for drag fores is onsidered. GMV douments fi the eentriit between and -3, but their simulations are obtained with 3 ε =. The inlination angle is 96.5 and the RAAN angle is Ω= 93 assuming a launhing date on tober 4, 4. The semi-maor ais is assumed to be the equatorial earth radius plus the satellite mean altitude, i.e. a = 668. km. Sine the perigee oinides with the equatorial radius, the line of nodes oinides with the diretion of the perigee, and ω = follows. The numerial values of the orbital parameters are summarized in Table. Dnamis_revised.do data reazione /7/ Pagina 7 di 8

18 parameter smbol value unit semi-maor ais a 668. km eentriit ε 3 inlination i 96.5 deg RAAN Ω 93 deg argument of the perigee ω deg Initial epoh t TBD s Table : numerial values of the orbital parameters The numerial values of the S/C mass and inertia parameters are summarized in Table. parameter value unit mass 9 kg 7 kgm 357 kgm zz 33 kgm Table : numerial values of the S/C mass and inertia parameters With these parameters, the initial position and veloit vetor omponents are: 587. r = 695. km.889 v = km/s 7.77 If the RAAN angle is assumed to be Ω= the initial onditions are 66.5 r = km v = km/s 7.77 (.6) (.63) Dnamis_revised.do data reazione /7/ Pagina 8 di 8