guidance, flight mechanics and trajectory optimization

Ii--
From this sketch, the following differential equations can be written for
the motion of the vehicle
and
*= - La+ u,
V
where T = the maximum thrust available
max
V = the exhaust velocity of the rocket.
(2.2.10)
(2.2.11)
(2.2.12)
There are several ways to formulate this problem for a Dynamic Programming
solution. The method used here is to represent the state of the vehicle by
four parameters, x, y, k, and y. The mass is used as a cost variable. The
four dimensional state space is divided into small intervals in each
coordinate direction. The coordinates designated by all the combinations
of various intervals form a set of points in the state space. Tne vehicle
starts at the initial point in the state space with some initial mass. The
control and mass change that are necessary to move the vehicle from this point
to the first allowable set of points in the state space are then computed.
This computation corresponds to the first set of possible control decisions.
Each end point of the set of possible first decisions is assigned a mass
(cost) and the path that gave the cost (for the first decision the path is
obvious since it must have come from the origin ).
The second decision is now investigated. The required control and the
correspondtig mass change required to go from the set of points at the end
of the first decision to the set of all possible points at the end of the
second decision must now be calculated. (The initial mass used jn this
second stage calculation is the mass remaining at the end of the first stage.
However, each point corresponding to the end of the second decision will
have more than one possible value of mass (depending on the point from which
it came). Thus, since it is desired to minimize the fuel consumed or
maximize the burnout mass, the largest mass is chosen as the optimum value
for that particular point. The point from which this optimum path came
is then recorded.
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