I thought I would ask what the most used methods in industry are, in order to guide a launch vehicle into a desired circular orbit at engine cutoff. It seems that the use of direction collocation methods in combination with a non-linear programming approach are common means of calculating optimal ascent trajectories, although they seem to be computationally taxing and thus cannot be used in real-time. Another approach that seems to be mentioned often in the literature is the linear-tangent steering (LTS) law which is a near-optimal guidance law developed in the 1960s. So my question is: Is the LTS method still the approach that is most often used for ascent guidance, or are there "more optimal" methods that can be used in real-time. Furthermore, how close to optimal is the LTS law? The literature often mentions the assumptions used in its derivation, but I have not managed to find any comparison of how well it compares to truly optimal solutions.
$\begingroup$ What is the linear-tangent steering law? Could you add a link within the text of the question? See for example how this question did it. It seems central to the question. This would improve the question's value to others (including me) who don't already recognize the term. Other examples might include this or this or this. Thanks! $\endgroup$– uhohApr 11, 2017 at 19:45
NASA Orion / SLS is still using PEG:
Note that "linear tangent steering" is a provably optimal guidance law from calculus of variations. It is going to be used in any finite-burn guidance program. What you're probably asking is if PEG is being used--which is a predictor-corrector method that uses analytical approximations instead of integration of trajectories (e.g. using RKF45 or whatever), which is fast enough to be used closed-loop.
Also note that guidance programs like PEG are not used for the atmospheric part of ascent. They're still used only used (AFAIK) exoatmospherically.
The ascent profiles for the in-atmosphere (endoatmospheric) are still calculated more laboriously on the ground. And a lot of mission planning is still done from the ground in plugged into PEG using the "external delta-V" mode. I don't know what is currently being used on the ground, in the late 60s it was the calculus of variations approach using runge-kutta integration and newton's method.
To confuse things slightly PEG's predictor could be replaced by discrete integration with Runge-Kutta which would improve accuracy and remove all the small-angle approximations and gravity integral approximations involved in PEG but at the expense of additional computation. I don't know what the current state-of-the-art is with the predictor and the gravity integral calculations with PEG.