1. Spacecraft Translational Motion
AOE
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. Pop Quiz W h a t r e p i n c l x s o f g u ? A w d - m W r i t e d o w
AOE W h a t r e p i n c l x s o f g u ? A w d - m W r i t e d o w n a l s f h g v u - m x : I = 2 4 8 1 7 6 3 5 c y ? H b

3. Pop Quiz (2) I = 2 4 5 1 3 T h e i g n v c t o r s a E ¡ : 8 9 7 6 W l
AOE I b = 2 4 5 1 3 T h e i g n v c t o r s a E : 8 9 7 6 W l ^ x ? .

4. Pop Quiz (3) A = 2 6 4 1 3 7 5 W h a t r e i g n v l u s ? k H o w d y
AOE A = 2 6 4 1 3 7 5 W h a t r e i g n v l u s ? k H o w d y c ( ) f m

5. Generalized Eigenvectors
AOE W h e n a m t r i x s d c g v l u , o p - f y [ 1 A ] = T I w ` ( ) T h e n u l s p a c o f [ ` 1 A ] m y t v d i , w r g I z C x - : ) R 2

6. Repeated Eigenvalues with Complete Set of Eigenvectors
AOE 2 4 p 3 1 5 h a s e i g n v l u ; = T m - t c y ` A r k , d o f : O b v i o u s l y e = [ ; 1 ] a n g c t r f A h p 2 T d m - 4

7. Repeated Eigenvalues with Complete Set of Eigenvectors (2)
AOE 2 4 p 3 1 5 E i g e n v a l u s r ; = c t o W f m h d x A S i n c e t h r s a o m p l f g v , d u y x E : A - . w b z Q = T

8. Repeated Eigenvalues with Incomplete Set of Eigenvectors
AOE 2 4 1 8 6 3 5 h a s c r t e i p o l y n m ( ) T g v u = - ` A k , d f : E l e m n t a r y o w d u c i s h g v f = 2 : [ 1 4 ] T z b p A S q

9. Repeated Eigenvalues with Incomplete Set of Eigenvectors (2)
AOE E l e m n t a r y o w d u c i : 1 = [ ] T 2 3 h s , x 4 5 f p A N e d t o k n w A f r h s b l c m : = E 1 2 4 3 5 R C . M a V L , \ i D u W y p x -
" S I v 9 7 8 { 6

10. Translational Motion Equations for a Satellite
AOE N e w 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

11. ~ r Orbital Frame P o s i t n c r u l a b : ~ = ¡ ^ e +
AOE
Orbital Frame
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

12. Translational Motion Equations for a Satellite
AOE T h 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

13. Translational Motion Equations for a Satellite
AOE C 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

14. Verify that Circular Orbit is an Equilibrium
AOE 2 4 Ä r 1 n _ 3 + 5 = F o a c i u l b t ( w h z e q ) , p s v - m ; d j S g f x y W e w i l n a r z b o u t h c s , p q d f m : _ x = ( ; )

15. g Standard Form for EOM 2 4 Ä r ¡ n _ + 5 = ¹ ¿ D e ¯ s i x t a , ( )
AOE 2 4 Ä r 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 +

16. Linearize About Circular Orbit
AOE T h 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 +

17. Linearize About Circular Orbit (2)
AOE C i r c u l a o b t : ( 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 Ã

18. Linearized System Near Circular Orbit
AOE
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)
Also, the system is in a decoupled form that is common for mechanical systems W h a t r e i g n v l u s o f A ?

19. Linearized System Is Decoupled
AOE
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

20. Linearized System Is Decoupled (2)
AOE T 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