guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
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performed using the Pontryagin Maximum Principle as in the last section.<br />
The notation used here is the same as in the previous monograph on the<br />
Pontryagin Maximum Principle (Reference 1.8). In order to make use of the<br />
maximum principle in the form given in the reference, Equation (4.36) must<br />
be reduced to a system of first order differential equations. This is<br />
accomplished by making the change of variables.<br />
The new equations of motion are:<br />
.<br />
or & = A & + Ba,. (4.37)<br />
The target set in which the terminal state must lie is given by:<br />
<strong>and</strong>, the function to be minimized is:<br />
The differential constraints are given by Equation (4.37); (in component<br />
form) they are:<br />
where Ai. J <strong>and</strong> Bj are elements of A <strong>and</strong> B, respectively,<br />
Notice that<br />
X,($~=O<br />
the target set 3v XF, tF 1 does not contain the condition<br />
. This condition is unnecessary since Xl (tf) can be nulled<br />
by advancing or delaying the time to start the rendezvous maneuver. If x1<br />
(t ) were constrained to be zero, then the problem would have to be refo$mulated<br />
to allow for coast periods, i.e., the acceleration should be<br />
allowed two values cl,&- , <strong>and</strong> 0 instead of just a, . In addition, the final<br />
time of the maneuver would no longer be proportional to the fuel used.<br />
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