guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
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Again, a test of the second derivatives will indicate that only the plus<br />
sign for D is allowed. The quadrant<br />
determined by the sign of C.<br />
for a (- 3 4 or L YZ ) is thus<br />
The portion of the Hamiltonian containing the control is<br />
H, = b (D + q) = b (D + p7)<br />
Thus, it is clear that to maximi ze the Hamiltonian when D + P7 '0, the<br />
value of b should be uo, whereas when D + P7 < 0, b should be 0. The case<br />
when D + P7 = 0 over an appreciable time is called a singular arc <strong>and</strong> is discussed<br />
by McIntyre (p 67 of Reference 4.12). Such arcs are not excluded <strong>and</strong><br />
have not been found to occur for these problems. Note that since D is either<br />
positive or zero, the value of P7 must be negative for coasting arcs to occur.<br />
The function k = D + P is called the switching function since it controls<br />
the value of the thrus z . Thus, to summarize<br />
If k Lo, fhen 6-O a coasting arc, or<br />
IT k 70, &nb=U, a full power thrusting arc<br />
<strong>and</strong> the thrust components are<br />
ul = b A/D<br />
u2 = b B/D<br />
u3 = b D/C.<br />
The distinction between the time optimal problem <strong>and</strong> the fuel optimal<br />
problem lies in the function to be optimized, the constraints, the boundary<br />
conditions on the adjoint variables, <strong>and</strong> the Hamiltonians. The time optimal<br />
problem is the determination of the controls such that<br />
4.27<br />
4.28<br />
4.29<br />
J = Qf isaminimum 4.30<br />
under the constraint that w7 ( Qf ) f AV, the total allowab:e velocity<br />
increment. It is necessary to include the constraint w (Q,) - A V in<br />
some way since otherwise P = 0 <strong>and</strong> the engine will be 7. urned on all the<br />
time. (That is, the time ?I o rendezvous would be minimized if there were<br />
no coast arc; that is,the vehicle should accelerate thus increasing the<br />
closing rate until it is necessary to reverse the thrust to be able to<br />
stop at the target). If the fuel is limited, this result, suggests that the<br />
optimal time will be attained by using all of the fuel <strong>and</strong> hence, w7 (Qf) =<br />
AV is the boundary condition. Thus, the seventh constraint is taken to be<br />
Y7 = “7 @,I - A V = 0 for the time optimal problem.<br />
65