design space pruning heuristics and global optimization method for ...

control is given by Equation 5, and consists of the thrust-direction unit vector, the thrust
magnitude, and the power. For a variable specific impulse trajectory, c, the exhaust
velocity, is a function of the jet power and thrust, as presented in Equation 6. For a
constant specific impulse trajectory, such as those used in the GTOC problems, the power
is not required as a control variable, and c is constant.
⎡ •
v ⎤ ⎡
•
⎤
⎢ r
X • •
⎥ ⎢ v
= ⎢ v ⎥ = − µ /r 3
) v ⎥
⎢ (
⎢
m • ⎥
( T /m )u)
⎥
⎢
⎥
⎣ ⎢ ⎦ ⎥ −( T /c)
⎣ ⎢
⎦ ⎥
(4)
⎡u⎤
u
⎢ ⎥
c
=
⎢
T
⎥
⎢⎣
P⎥⎦
(5)
Tc
P = (6)
2
For each leg of the trajectory, the spacecraft’s initial conditions are determined by
the position and velocity of the departure body at a specified time. At rendezvous, the
spacecraft must also match the position and velocity of the target body. The final time, t f ,
may be fixed or free, depending on the problem formulation. These terminal state
constraints are given in Equation 7.
⎡
C = r s/ c
⎢
⎣ ⎢ v s/ c
( )
⎤
⎥
()− v t ()
()− t f
r t
t f
t f
t f
⎦ ⎥ = ⎡ 0 ⎤
⎢ ⎥ (7)
⎣ 0⎦
Finally, there are additional constraints on the maximum thrust and power, as specified
by the chosen spacecraft and engine parameters.
In general, there are two types of methods for solving the local trajectory
optimization problem – direct and indirect. 12,13,14,15
Indirect methods are based on
5