Nathan was asking this evening what we collectively know about
computing an optimal orbital-insertion trajectory, and sharing giant
1960s papers on the subject that were written using typewriters.
(Nathan, if those papers are available publicly, could you post links
please?)

So I started trying to list the pieces that I think I understand. I'd
appreciate comments. I think I'm just re-deriving the "basics" but
they're not obvious to me, so maybe this is useful to others too?

The goal, as I understand it, is to minimize total fuel consumed, with
a constraint on the maximum thrust at any instant. We have some
initial and final boundary conditions, and a variety of pre-specified
constants: payload mass, motor Isp, drag coefficients, etc.

For correct results we also need to account for the non-fuel mass of
each stage's motor casings, but I'd be pretty happy if we had a model
that gave optimal answers assuming zero-mass motor casings. You can
get a conservative estimate from such a model by folding the total
motor casing mass into the payload mass; then you've modeled a
single-stage rocket.

The initial conditions are a known velocity and position. (This works
for plane or balloon launches too, I guess?) The final conditions are
a pre-specified orbital altitude, and speed in a direction
perpendicular to gravity, and a fixed remaining fuel mass. (I believe
the final position over ground must not be fixed.)

I think the forces that matter are drag, thrust, and gravity. If angle
of attack should be zero or near enough for the bulk of the orbital
insertion, then I think we can ignore the effects of a non-zero angle
of attack. I also assume that forces due to maneuvering are small
compared to everything else, and can also be ignored. Perhaps these
are over-simplifications?

I assume the total fuel consumed is proportional to the integral of
the thrust force. Wikipedia seems to be telling me that I've just
given a definition for "thrust specific fuel consumption", which is
inverse to specific impulse (Isp), so I think that's right.

Gravity gives us an acceleration in only the vertical direction and
dependent only on altitude, not mass. Thrust and drag are opposite
forces that both must be divided by the mass, which is related to the
integral of thrust; and further, drag depends both on altitude (by way
of air pressure) and on velocity. I never took differential equations,
but... these are, and they're non-linear.

Our control variables are thrust and orientation. I think it's easy to
write down the partial derivatives we're interested in given those
control variables as a function of time, but my notation is weak. So
of course it's left as a trivial exercise for the reader.

Sombody this evening mentioned calculus of variations, which I didn't
know anything about, but sure enough both it and optimal control
theory are totally relevant here. I'll conclude with some links:

http://en.wikipedia.org/wiki/Calculus_of_variations
http://en.wikipedia.org/wiki/Optimal_control
http://en.wikipedia.org/wiki/Boundary_value_problem
http://en.wikipedia.org/wiki/Nonlinear_programming

The optimal control article mentions OTIS, by the way.

Jamey

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