guidance, flight mechanics and trajectory optimization

time the vehicle crosses the desired plane. This cost is
For the finite thrust case of (m+l) intervals of acceleration (u3) and
duration @,-the cost is
This technique is shown to satisfy the Pontryagin Maximum Principle by the
development of the switching function.
For the in-plane motion Tschauner and Hempel point out that
This equation shows that 7, (0) is a lower limit for the velocity increment
needed, and that this minimum can be reached only if there is no radial thrust
and the circumferential thrust does not involve a change in direction. On
the basis of this argument, ul is set = 0 and the problem is reduced to a
one dimensional motion.
The condition that ~2 does not suffer a change in direction is known
to occur when the two orbits do not intersect, since for this case optimum
impulsive transfer is very nearly cotangential, and the angular momentum is
increased by both burns (the active vehicle is assumed to be in the orbit
closer to the force center). Tschauner and Hempel proceed to develop cri-
teria and equations for a three burn maneuver of the form shown in Figure 4.1.
-
CIRCuM. -R - - @2 -
ACCEL.
1 J
Figure 4.1. Three-Burn Maneuver to Rendezvous
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4.3
4.4