( ) R O , i , j , k is an Earth centered equatorial - PDF document

Dynamic Model of the Spacecraft Position and Attitude Basilio BONA, Enrico CANUTO Dipartimento di Automatica e Informatica, Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino, Italy tel. 011 564 7026, fax 011 564 7099 bona@polito.it 1 Reference Frames 1.1 Reference Frames Description In this report the following reference frames will be considered: 1.1.1 Inertial Reference Frame – IRF ( ) R O , i , j , k is an Earth centered equatorial inertial frame, with: J2000 J2000 J2000 J2000 J2000 O � at the centre of the Earth. J2000 i � along the intersection of the mean ecliptic plane with the mean equatorial plane, at the date of J2000 01/01/2000; positive direction is towards the vernal equinox. k � orthogonal to the mean equatorial plane, at the date of 01/01/2000; positive direction is J2000 towards the north. j � completes the reference frame. J2000 1.1.2 Earth-fixed Reference Frame – EFRF ( ) R O , i , j , k is an Earth fixed non-inertial frame, and shall be the most recently defined E E E E E International Terrestrial Reference Frame (ITRF). We assume: O at the centre of the Earth. � E i along the intersection of the equatorial plane and the Greenwich meridian plane; positive � E direction from the earth centre to the 0° longitude point on the equator. k orthogonal to the equatorial plane; positive direction towards the north pole. � E j completes the reference frame. � E 1.1.3 Orbit Plane Reference Frame – OPRF ( ) R O , i , j , k is a local non-inertial orbit plane reference frame, with: OP OP OP OP OP O � at the centre of the Earth. OP i � along the intersection of the orbit plane with the equatorial plane. OP k k positively around OP i by the orbit inclination angle i . � obtained by rotating OP E Dynamics_revised.doc data creazione 12/ 07/ 2001 9.35.00 Pagina 1 di 18 data ultima revisione: 29/ 04/ 2002 8.45.00

j � completes the reference frame. OP 1.1.4 Local Orbital Reference Frame – LORF ( ) R O , i , j , k is a non-inertial local orbital reference frame, with: O O O O O O at the actual satellite centre of mass. � O i (roll) parallel to the instantaneous direction of the orbital velocity vector v = � r ; positive � O direction towards positive velocity; O = v i . v j (pitch) parallel to the instantaneous direction of the orbital angular momentum h = r × v , where � O × r v r is the vector from the Earth centre to O ; j = . O O × r v × v h k (yaw) completes the reference frame: k = = i × j � . O O O O × v h 1.1.5 Spacecraft Reference Frame – SCRF ( ) R O , i , j , k is a local satellite non-inertial frame, with: SC SC SC SC SC O � at the centre of the launcher-satellite adaptor interface plane SC i � along the launch vehicle axis; positive direction is towards the launch vehicle nose. SC k � orthogonal to the satellite earth face; positive direction is towards nadir. SC j � completes the reference frame. SC 1.1.6 Attitude Control RF – ACRF ( ) R O , i , j , k is a local non-inertial satellite reference frame, with: AC AC AC AC AC O � at the actual satellite centre of mass (COM). AC i , j , k i , j , k � parallel to SC . AC AC AC SC SC 1.2 Transformation Matrices between Reference Frames ( ) ( ) ( ) α β γ R i , , R j , , R k , In the following we use the notations to represent, respectively, rotations around the X , Y , and Z axis. We remember that a rotation about fixed axes pre-multiplies, while a rotation about the moving axes post-multiply. ⇒ J R R R a) , given by matrix : J2000 OP OP ⎡ ⎤ − c s c s s ⎢ ⎥ Ω Ω Ω i i ( ) ( ) ⎢ ⎥ J = Ω = − R R k , R i , i s c c c s (1.1) ⎢ ⎥ Ω Ω Ω OP i i ⎢ ⎥ 0 s c ⎢ ⎥ ⎣ ⎦ i i Dynamics_revised.doc data creazione 12/ 07/ 2001 9.35.00 Pagina 2 di 18 data ultima revisione: 29/ 04/ 2002 8.45.00

notice that, while the inclination angle i is constant (see Table 2), the right ascension Ω varies at a ⋅ -7 1.996425 10 rate of 0.9856 deg/day ( rad/s), to allow the satellite to be sun-syncronous ⇒ J R R R : b) , given by matrix J2000 E E ⎡ ⎤ − c s 0 ⎢ ⎥ δ δ ( ) ⎢ ⎥ J = δ = ⎢ R R k , s c 0 (1.2) ⎥ δ δ E ⎢ ⎥ 0 0 1 ⎢ ⎥ ⎣ ⎦ notice that the angle δ varies at a rate of 360 deg/day ( ⋅ -5 7.2921 10 rad/s), due to daily rotation. ⇒ J R R R : c) , given by matrix J2000 O O ⎡ ⎤ v r × v v × × r v ⎢ ⎥ J = ⎢ R (1.3) ⎥ O ⎢ v r × v v × × r v ⎥ ⎣ ⎦ where v and r are defined in Section 1.1.4 . R ⇒ R E R d) , given by matrix : E OP OP ′ ( ) E = E J = J J = R R R R R OP J OP E OP ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ − + − + − c s 0 c s c s s c c s s c s c s c c c s s s c s ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1.4) δ δ Ω Ω i Ω i δ Ω δ Ω δ Ω i δ Ω i δ Ω i δ Ω i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − − = − + + − − s c 0 s c c c s s c c s s s c c c c s s s c c s ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ δ δ Ω Ω i Ω i δ Ω δ Ω δ Ω i δ Ω i δ Ω i δ Ω i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 1 0 s c 0 s c ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ i i i i R ⇒ R R OP e) , given by matrix : OP O O ⎡ ⎤ − s 0 c ⎢ ⎥ α α ( ) ( ) ⎢ ⎥ OP = � � + α = ⎢ R R i , 90 R j , 90 c 0 s (1.5) ⎥ O α α ⎢ ⎥ 0 1 0 ⎢ ⎥ ⎣ ⎦ where α is the true anomaly. R ⇒ R R O f) , given by matrix : O AC AC ⎡ ⎤ c c c s s − s c c s c + s s ⎢ ⎥ ψ θ ψ θ ϕ ψ ϕ ψ θ ϕ ψ ϕ ( ) ( ) ( ) ⎢ ⎥ O R = R k , ψ R j , θ R i , ϕ = s c s s s + c c s s c − c s (1.6) ⎢ ⎥ AC ψ θ ψ θ ϕ ψ ϕ ψ θ ϕ ψ ϕ ⎢ ⎥ − s c s c c ⎢ ⎥ ⎣ ⎦ θ θ ϕ θ ϕ where ϕ = roll angle, θ = pitch angle, ψ = yaw angle; for small angles the above matrix reduces to: ⎡ ⎤ − ψ θ 1 ⎢ ⎥ ⎢ ⎥ O = ψ − ϕ R 1 (1.7) ⎢ ⎥ AC ⎢ ⎥ − θ ϕ 1 ⎢ ⎥ ⎣ ⎦ Dynamics_revised.doc data creazione 12/ 07/ 2001 9.35.00 Pagina 3 di 18 data ultima revisione: 29/ 04/ 2002 8.45.00

2 Dynamic State Space Model 2.1 Assumptions The state space model derived in this report describes R a) the dynamics of the S/C COM position with respect to the inertial frame ( orbital dynamics ); J2000 R R b) the dynamics of the S/C reference frame with respect to the orbital frame ( attitude AC O dynamics ). The following assumptions hold: R Assumption 1 : the dynamics of the COM position is described by the differential equations of the AC z ′ ⎡ ⎤ R r = ⎢ x y origin represented in by the vector . The COM velocity origin is represented in ⎥ ⎣ ⎦ J2000 v ′ ⎡ ⎤ R v = ⎢ v v by the vector . ⎥ ⎣ ⎦ J2000 x y z R R Assumption 2 : the attitude of with respect to is given by the three roll-pitch-yaw angles AC O ψ ′ ⎡ ⎤ α = ϕ θ ⎦ implicitely defined in eqn. (1.6). ⎢ ⎥ ⎣ R Assumption 3 : in order to transform the vectors represented in a generic referecne frame into vectors A R represented in another reference frame , the following relation holds: B ⎡ ⎤ ⎡ ⎤ B p = R p (1.8) ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ A R R B A ⎡ ⎤ B p R R is a orthonormal matrix with where represents a generic vector in the reference frame and ⎢ ⎥ ⎣ ⎦ R B A B positive unitary determinant (proper rotation). Assumption 4 : inputs and/or measurements, defined or obtained in reference frames other than the reference R frame , are assumed to be already represented in this reference frame. AC i j k , , Assumption 5 : the axes of the A/C reference frame are assumed to coincide with the principal AC AC AC axes of inertia. Assumption 6 : the S/C mass m and inertia matrix ⎡ ⎤ J 0 0 ⎢ ⎥ x ⎢ ⎥ = ⎢ J 0 J 0 (1.9) ⎥ y ⎢ ⎥ 0 0 J ⎢ ⎥ ⎣ ⎦ z are assumed to be time-invariant, although this is not true, due to propellant consumption. 2.2 Plant Dynamic Equations In order to model the S/C dynamics it is necessary to introduce first the orbital kinematic equations. 2.2.1 Kinematic equations Assumption 7 . The total inertial velocity vector ω is the sum of the orbital rotation velocity and the S/C attitude rate. Dynamics_revised.doc data creazione 12/ 07/ 2001 9.35.00 Pagina 4 di 18 data ultima revisione: 29/ 04/ 2002 8.45.00