guidance, flight mechanics and trajectory optimization

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