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Let's consider the Fixed Arrival Time problem.
We want to find a minimum fuel solution that takes our craft from to starting (x, y)
position to the target (x, y) position so that it arrives at (exactly) time T and we want to
get there in N time steps each of time
∆
t.
T/N=
∆
t.
There is no notion of negative fuel so
we can instead think in terms of thrust in four dimensions ForwardThurstX (ft
x
),
BackwardThrustX(bt
x
), ForwardThrustY(ft
y
) and BackwardThrustY(bt
y
).
This is by no
means the only way of representing this information—it is just an example designed to be
easy to understand.
Therefore our objective function is to minimize J
T
as follows:
∑
−
−
=
1
0
min
min
N
i
i
T
U
U
U
Cost
i
i
We will assume that thrust in each direction burns the same amount of fuel so:
Cost
T
=
()
1
1
1
1
Where U
i
=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
i
i
i
i
bty
fty
btx
ftx
We are solving to get a sequence of N states S
0
… S
N1
resulting from a sequence of N
control inputs U
0
… U
N1
.
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This note was uploaded on 11/07/2011 for the course AERO 16.410 taught by Professor Brianwilliams during the Fall '05 term at MIT.
 Fall '05
 BrianWilliams

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