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.
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.