1. Spacecraft Translational Motion
The two-body problem involves the motion of a satellite caused by the gravitational attraction of a central body
The circular orbit solution is well-known and is useful for many satellite applications
This analysis is about the translational motion of a satellite “close to” a circular orbit

2. Translational Motion Equations for a Satellitew t o n : m Ä ~ r = G M 3 + T h s a l i c b d y p v g u D , T h e c o n s t a , G M i u l y d A f r m b F ( H ) w g v ~ ! = ^ 2 p 3 - j x

3. ~ r Orbital Frame P o s i t n c r u l a b : ~ = ¡ ^ e +
Same as “roll-pitch-yaw” frame, for spacecraft
The o3 axis is in the nadir direction
The o2 axis is in the negative orbit normal direction
The o1 axis completes the triad, and is in the velocity vector direction for circular orbits ~ r P o s i t n c r u l a b : ~ = ^ 3 e 1 + 2

4. Translational Motion Equations for a Satelliteh e v c t o r q u a i n f m - : Ä ~ = G M 3 + F s p , [ 1 2 ] ! I l w y b d T h e v l o c i t y r a s m - p n , g b d = _ + ! C u 2 4 1 3 5

5. Translational Motion Equations for a SatelliteC a r y i n g o u t h e p l d s 2 = 4 Ä 1 _ 3 + 5 N w c q f m : T h e t r s c o n d - l i a q u b p f m , w j _ = 1 ; 2 3 C

6. Verify that Circular Orbit is an Equilibrium2 4 Ä r 1 n _ 3 + 5 = F o a c i u l b t ( w h z e f ) , p s v - m ; d j S g x y q L i n e a r z b o u t h c l s , p q d f m : _ x = ( ; )

7. g Standard Form for EOM 2 4 Ä r ¡ n _ + 5 = ¹ ¿ D e ¯ s i x t a , ( )1 n _ 3 + 5 = D e s i x t a j , 6 ( ) g T h e s q u a t i o n r d f m : _ x = ( ; ) w , 3 2 1 +

8. Linearize About Circular Orbith e r s t f u n c i o a y d ; . g , @ x j = + 3 1 2 l k C i r c u l a o b t : ( x ; ) = [ ] _ 1 4 2 5 3 6 n +

9. Linearize About Circular Orbit (2): ( x ; ) = [ ] _ 4 n 2 3 1 + 6 5 S e v m s f j . h p - , d k w @ E v a l u t i n g h e c r - o b m s x 1 = 2 , 3 4 5 6 Ã

10. Linearized System Near Circular Orbit
Completing the partial derivatives and simplifying leads to: _ x = 2 6 4 1 n 3 7 5 + u A B
Clearly this system is controllable and stabilizable (exercise) W h a t r e i g n v l u s o f A ?

11. Linearized System Is Decoupled
Two rows and columns are decoupled from the rest: _ x = 2 6 4 1 n 3 7 5 + u A B
These two states are decoupled from the rest and so the system can be arranged into a 22 system and a 44 system

12. Linearized System Is Decoupled (2)h e 2 s y t m i n v o l x a d 5 , w c r b u - f p : _ = 1 + 4 3 ) g ( z W P I D k

13. Out-of-Plane Motion T h e 2 £ s y t m i n v o l x a d , w c r b u - f5 , w c r b u - f p : _ = 1 + 4 3 ) q Ä I 6 [ ; ( ]

14. Out-of-Plane Motion (2)h e o u t - f p l a n m r i x s : A = 1 2 g v , d c E b k z )

15. Out-of-Plane Motion (3)h e b l o c k - d i a g n z t r s f m x E = 1 p u ( ) + 2

16. In-Plane Motion T h e 4 £ s y t m i n v o l x ; , w c d r b - p a ( k1 ; 3 6 , w c d r b - p a ( k ) : _ = 2 7 5 + u ` F g C f [ ] A z . S G q

17. In-Plane Motion (2) T h e 4 £ s y t m a p l n r i x : A = 2 6 1 3 ¡ 71 3 7 5 w ; ( ` ) F o , [ ] d g v c f E N b k - .

18. In-Plane Motion (3) T h e s t a r n f o m i x E = 2 6 4 1 ¡ ( 3 ) 7 5( 3 ) 7 5 p l c d u y

19. In-Plane Motion (4) T h e s t a r n i o m x ( E c ) = 2 6 4 1 ¡ + 7 53 + 7 5 M b g w l v u d . k p y f , -

20. In-Plane Motion (5) T h e 4 £ s y t m i n v o l x ; , w c d r b - p a1 ; 3 6 , w c d r b - p a : _ = 2 7 5 + u C E D ( z f )