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.

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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/Max_Q http://en.wikipedia.org/wiki/Gravity_drag http://en.wikipedia.org/wiki/Gravity_turn 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 _______________________________________________ psas-team mailing list psas-team@lists.psas.pdx.edu http://lists.psas.pdx.edu/mailman/listinfo/psas-team This list's membership is automatically generated from the memberships of the psas-airframe, psas-avionics, and psas-general mail lists. Visit http://lists.psas.pdx.edu to individually subscribe/unsubscribe yourself from these lists.