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

Translational Motion Equations for a Satellite 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 ¹ A f r m b F ( H ) w g v ~ ! = ¡ ^ 2 p 3 - j x

~ 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

Translational Motion Equations for a Satellite 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

Translational Motion Equations for a Satellite 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 : ¹ ¿ 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

Verify that Circular Orbit is an Equilibrium 2 4 Ä r 1 ¡ n _ 3 + 5 = ¹ ¿ 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 = ( ; )

g Standard Form for EOM 2 4 Ä r ¡ n _ + 5 = ¹ ¿ D e ¯ s i x t a , ( ) 1 ¡ n _ 3 + 5 = ¹ ¿ 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 +

Linearize About Circular Orbit 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 ¹ + ¿

Linearize About Circular Orbit (2) : ( x ¤ ; ) = [ ¡ ] _ 4 n 2 ¹ 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 ¹ Ã

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 ?

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

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

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 - f 5 , w c r b u - f p : _ = · 1 ¡ ¸ + 4 3 ) § q Ä I 6 [ ; ( ]

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 ¤ )

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

In-Plane Motion T h e 4 £ s y t m i n v o l x ; , w c d r b - p a ( k 1 ; 3 6 , w c d r b - p a ( k ) : _ = 2 ¡ 7 5 + · u ¸ § ` F g C f [ ] A ¯ z . S ° G q

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

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

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

In-Plane Motion (5) T h e 4 £ s y t m i n v o l x ; , w c d r b - p a 1 ; 3 6 , w c d r b - p a : _ = 2 ¡ 7 5 + · u ¸ C E D ( z f )