On Mar 9, 2011, at 3:22 AM, Jamey Sharp wrote:

> 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.

Another constraint is to minimize maximum dynamic pressure (max Q), a function 
of velocity and altitude.  This is why we don't launch bulk items into orbit 
from cannons.  It is also a reason why launch trajectories tend to punch up 
through the dense atmosphere before turning, despite gravity drag. Even air 
launched rockets turn *up* after their initial horizontal launch.

More links:

http://en.wikipedia.org/wiki/Trajectory_optimization   (lists five programs, 
including OTIS)

> 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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