guidance, flight mechanics and trajectory optimization

Thus in cylindrical coordinates, the position of the target satellite is
(r' , 0, 0) and its motion satisfies the differential equation (the ' indica+es
d/dt)
while the active
satisfies
where a is the
ing rerZezvous.
p
the two satellites and 'is given by
/i’ = - pJp g,
--I
satellite is at i&(1 t [,I, B + e2, rl t31 and its motion
instantaneous acceleration produced by the engines in affect-
The differential equation for the relative position e= r - 3 is then
Agt 5 where Ag is the difference in gravitational accelerations of
Equation 1.6 may have a wider application than indicated here, for Ag may
represent the difference in all accelerations of the two satellites. Thus,
the reference trajectory could be a simple orbit satisfying any portion of the
total force equation. In fact, the idea of referring the motion of one satel-
lite to that of another nearby has been used in lunar theory since the time of
Euler in 1772; and it is doubtful if any of the sets of differential equations
given below could be considered to be original in this century.
The signs of the coriolis terms in equations found in the literature are
sometimes the opposites of those used in this monograph. The difference arises
from a difference in the choice of coordinate axes, here X radial, Y cir-
cumferential ahead; whereas many authors use X circumferential back, Y
radial.
2.1.2 Field-Free Case
For the field free case, the vector difference Ag is assumed to be
negligibly small and one obtains simply $ = 2 The solutions are immediately
available as
6
0
1.4
1.6
1.7