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