References on Launch Trajectories

Are there any good texts/papers about computing trajectories from launch to LEO? Specifically discussing atmospheric effects, gravity turns, launch sties, etc. and numerical simulation. I've seen a lot on astrodynamics but not so much on actually launching the vehicle from the ground up.

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Dec 12, 2023 at 20:12

Most real rockets use a combination of two methods to guide their ascent. The first stage through the atmosphere uses "open-loop" mode, where it just steers according to a pitch program -- IE hit a pitch of 80deg 15s after launch, 70deg 30s after, etc. At first staging, the performance of the rocket changes radically. Also, at staging, most orbital launch vehicles are above the vast majority of the atmosphere and can continue steering without regard to keeping the pointy end into the wind. So, most rockets shift guidance programs as well as propulsion hardware at this point, using some "closed-loop" guidance mode. In this mode, a trajectory from the current position and velocity to the target position and velocity is continuously updated. Any errors in trajectory up to this point are considered, and the guidance program dynamically re-calculates its pitch program all the way to cutoff. It does this again and again (IE once every 2 seconds on Saturn or Shuttle) until it reaches the target orbit, steering out any errors. This kind of closed-loop guidance is needed to achieve any kind of orbit injection accuracy.

My favorite reference is "Explicit guidance equations for multistage boost trajectories" by Fred Teren. This doesn't cover all of the launch, instead focusing on second and third stage guidance. The reference problem was using an Atlas-Centaur to throw a Surveyor spacecraft onto an impact trajectory with the Moon. The algorithm covers from closed-loop guidance start after dropping the Atlas booster engines. It proceeds through dropping the Atlas sustainer, lighting the Centaur booster, and all the way to Centaur cutoff in a single burn targeting the lunar transfer trajectory. Notably it skips the parking-orbit phase.

While the algorithm is designed with this lunar trajectory in mind, it can be used to hit any altitude, horizontal speed, and vertical speed at burnout (providing the launch vehicle can actually achieve it). I have used it successfully to write an autopilot for a simulated Space Shuttle launch, which ends at low Earth orbit.

Another one from NTRS which is on my electronic shelf but I haven't actually read is "Guidance, Flight Mechanics, and Trajectory Optimization Volume 8 - Boost Guidance Equations"

"Spacecraft Navigation and Guidance" by Maxwell Noton includes a section on the launch phase, including gravity turn, drag, etc. I somehow got a PDF of this book, but I don't remember where. I originally used a hardcopy from my university library. If I recall, it wasn't super-useful because it wasn't closed-loop. You aimed it by firing with some particular tilt parameters, seeing where you ended up, then adjusting the tilt parameters and firing again and again until the trajectory hit the target orbit. Having said that, I think that's how real rockets are aimed. On the day of launch but before launch, the launch team runs a bunch of simulations using the currently observed wind and weather. Once the team finds a solution which safely gets the first stage through the atmosphere, it uploads that program to the launch vehicle which flies it on blind faith. Once the first-stage ends, the rocket uses its upper-stage guidance to hit the orbital target.

Some other useful search terms are "pitch program", "linear tangent guidance", and "iterative guidance mode" (the name of the algorithm that the Saturn V used in-flight).

Try:

"Introduction To Space Dynamics" by William Tyrrell Thomson

and

"Orbital Mechanics for Engineering Students" by Howard D. Curtis

I did search a while back for a good article on how to optimize a gravity turn, but I just didn't find anything that I found to be particularly earth-shattering (pardon the pun). In the end I just implemented my own iterative optimization approach.

But if you want to continue the literature search, it might help to include terms like "roll program" and "tilt maneuver".