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guidance, flight mechanics and trajectory optimization

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Note that if it, can occur, this scheme will include the optimal two-burn<br />

case when the thrust is not reversed.<br />

The integrals 21, z2, z3, <strong>and</strong> z can be easily found <strong>and</strong> set equal<br />

to wl (0), w2 (0), w3 (0), w4 (0). They are:<br />

..--._--_-._-_._ - _...-<br />

Assuming that Qf = 73 is given, it is seen that there are four equations<br />

to determine the unknowns #l, $2, $3, rl, <strong>and</strong> r2. Since the fuel consump-<br />

tion is equal to ($1 + @2 + a;), this sum can be equated to some reasonable<br />

value as the extra equation.<br />

The inversion of these equations to find the coast <strong>and</strong> burn intervals is<br />

not possible in a closed form, but by means of special assumptions Tschauner<br />

<strong>and</strong> Hempel are able to solve the special case of optimal rendezvous from an<br />

inner non-intersecting orbit. The results are a generalization from impul-<br />

sive to finite thrust of the impulsive splitting technique suggested in<br />

Ref. 4.6 <strong>and</strong> 4.7.<br />

Before leaving the work of Tschauner, it is to be noted that in<br />

Reference 1.6, a similar pair of thrusting programs for elliptic reference<br />

orbits for the out-of-plane <strong>and</strong> the in-plane portions of a rendezvous with<br />

a target in an elliptic orbit is developed. The independent variable for<br />

this analysis was eccentric anomaly (the analysis turns out to be mDre<br />

manageable). Again, no radial thrusting is assumed <strong>and</strong> again, the case of<br />

three impulses to rendezvous is developed. The analysis is somewhat involved<br />

<strong>and</strong> the reader is referred to the paper for thedetails <strong>and</strong> the equations.<br />

60

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