[PSAS] optimal trajectory dynamics
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 ___ 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.
Re: [PSAS] optimal trajectory dynamics
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?) All 17 volumes are here: http://naca.larc.nasa.gov/search.jsp?N=0Ntk=allNtx=mode%20matchallNtt=%22Guidance,%20flight%20mechanics%20and%20trajectory%20optimization%22 http://naca.larc.nasa.gov/search.jsp?N=0Ntk=allNtx=mode%20matchallNtt=%22Guidance,%20flight%20mechanics%20and%20trajectory%20optimization%22 Part of the reason it's so long is that it's written from base principles. You really don't need to read all of it, but many of the volumes are quite interesting. Here is the index of volumes: 1. Coordinate Systems and Time Measure 2. Observation Theory and Sensors 3. The Two Body Problem 4. The Calculus of Variations 5. State Determination and/or Estimation 6. The N-Body Problem and Special Perturbation Techniques 7. The Pontryagin Maximum Principle 8. Boost Guidance Equations 9. General Perturbations Theory 10. Dynamic Programming 11. Guidance Equations for Orbital Operations 12. Relative Motion, Guidance Equations for Terminal Rendezvous 13. Numerical Optimization Methods 14. Entry Guidance Equations 15. Application of Optimization Techniques 16. Mission Constraints and Trajectory Interfaces 17. Guidance System Performance Analysis The most interesting one to me is #15: Application of Optimization Techniqueshttp://naca.larc.nasa.gov/search.jsp?R=350776id=7as=falseor=falseqs=No%3D10%26Ntt%3D%2522Guidance%252c%2Bflight%2Bmechanics%2Band%2Btrajectory%2Boptimization%2522%26Ntk%3Dall%26Ntx%3Dmode%2Bmatchall%26Ns%3DHarvestDate%257c1%26N%3D0. It sums up the previous volumes and actually solves some simple optimization problems. -Nathan ___ 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.
[PSAS] imagine this: a joke
A 747's pilot and copilot both have heart attacks mid-flight. A flight attendant gets on the PA to ask if anyone knows how to fly a plane. Two Polish men stand up, saying that they're crop dusters. The flight attendant escorts them to the cockpit. They stare at all the levers and buttons in clear befuddlement, until one turns to the flight attendant and says, I'm sorry. We're just two simple poles in a complex plane. ___ 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.
Re: [PSAS] imagine this: a joke
Ben, You suck. Thanks for wasting 2 minutes of my life which I will never get back. Rob -- From: Benjamin Kaplin ben...@gmail.com Sent: Wednesday, March 09, 2011 4:24 PM To: psas-...@psas.pdx.edu Subject: [PSAS] imagine this: a joke A 747's pilot and copilot both have heart attacks mid-flight. A flight attendant gets on the PA to ask if anyone knows how to fly a plane. Two Polish men stand up, saying that they're crop dusters. The flight attendant escorts them to the cockpit. They stare at all the levers and buttons in clear befuddlement, until one turns to the flight attendant and says, I'm sorry. We're just two simple poles in a complex plane. ___ 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. ___ 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.
