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Similar Question: What computer and software is used by the Falcon 9?

What would the formula (function) calculating the changes in angle of attack of the Grid Fins (Falcon 9) before landing look like?

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This question highlights a newly developed tendency in the progress of Space Exploration. Long stagnated advances in the development of space rocket technologies (SLS as an example) were boosted by advances in computing power and Artificial Intelligence (SpaceX Falcon 9, Starship).

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Advances in onboard computers processing power and the 'brute force' approach to calculation of avionics control commands could have made vertical landing of the first stage practical and affordable for commercial entities like SpaceX.

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    $\begingroup$ Sarcastic answer is to point to 'the missile knows where it isn't' youtube.com/watch?v=c-FT0T9Ei-4 which was already a meme for being a terrible explanation for this in the 90s. The core remains the same though, with a known position and an aim point generate controls signals to reduce error. $\endgroup$
    – GremlinWranger
    Feb 18 at 22:51
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    $\begingroup$ Is it such a Giant Leap, though? Blue Origin also autonomously lands their New Shepard booster, and even did it successfully before SpaceX did (although, of course, New Shepard is not an orbital system, it just goes straight up and down again). The DC-X demonstrated autonomous vertical landing in 1993 (it was intended to land from orbit, but all demonstrations were just hops of at most ~3km; on its penultimate flight, it even demonstrated the Starship-style reorientation from belly-down to landing feets-down). Autonomous landing from orbit (but not vertical) was demonstrated with Buran in … $\endgroup$
    – Jörg W Mittag
    Feb 19 at 14:52
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    $\begingroup$ Have a look at bps.space . He did it with solid rocket motors and most of the stuff is public. $\endgroup$
    – Greg Miller
    Feb 19 at 23:36
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    $\begingroup$ ...doesn't quite satisfy that. If you hover a cursor over the up and down voting arrows, you can see that the tooltip even suggests prior research is a basis for voting up or down on questions. i.stack.imgur.com/rCROe.png We generally avoid making significant changes to questions once answers start to be posted to avoid making them a "moving target" and making current answers less valid. So in this case I think you can ask a follow-up reference-request question, perhaps something like: $\endgroup$
    – uhoh
    Feb 20 at 0:16
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    $\begingroup$ @TheMatrixEquation-balance ya that's the number-one challenge to get used to, having questions closed when we think they can have good answers. It was a real struggle for me in the beginning, but I found that if I deferred to the community a bit and kept at it (continued to post) I started getting better at constructing questions in the "Stack Exchange" way. The site is open to (almost) the entire internet, and yet is incredibly smooth and peaceful compared to most of it, and that's because it's fairly rigidly structured and the moderators and community do a good job of helping us adhere to it $\endgroup$
    – uhoh
    Feb 20 at 1:19

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While SpaceX has not published anything specific on the flight control logic, the underlying process is by no means magic.

The basic control mode for a system is generate a control signal that moves towards the set point. The most basic being the classic thermostat that turns off when too hot and on when too cold and oscillates either side. More precise control can be done with proportional control, where if far from setpoint you drive harder towards it. This has a tendency to drive hard and overshoot, so you also factor in rate of approach to set point and damp the drive rate down as it approaches.

This system is referred to as proportional-integeral-derivative (PID) control and are a well established technology. Where the magic happens is tuning the three values to achieve fast control without overshoot and/or oscillation. This is complicated for spacecraft descent in that the control effects change with speed and air density, so ideally you do a combination of modelling and testing to find ideal values for all possible speed/altitude/mass combinations.

This answers the question as written, but possibly not the question as intended in terms of how to actually determine the intended flight path. The SpaceX method does not appear to have been made public, but published work by others seem to use a combination of pre computed path switching to a live terminal descent mode that predicts impact point without control inputs and iterates towards a sequence of control inputs that will achieve touchdown. The involved maths is complex for a human but solvable even with Apollo era hardware, though faster processing allows more optimal solutions to be found without impacting during number crunching or by exceeding stability limits. In particular Apollo Lunar descent had a rather simpler inverted pendulum to deal with than SpaceX.

Powered descent has even been achieved by a hobbist on a solid rocket motor which while working only at a single atmospheric pressure and without aerodynamic surfaces is of interest in showing the mechanical and PID tuning process involved.

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Landing of the Falcon 9 is very challenging from a number of perspectives. Firstly it can't hover as the engine thrust cannot be throttled sufficiently to prevent it accelerating into the air. So the only option is to ensure that deceleration to 0m/s is matched to 0m altitude and then turn off the engine.

As for the grid fins I suspect that initially they made some best guess estimates based on simulation of what should happen to the rocket if the grid fins moved in a particular way and then tried it out with a lot of telemetry. From that they must have discovered exactly how the grid fins do affect the flight profile when they move in a particular way.

It took them a considerable time and lots of exploding boosters, but they eventually managed to iterate the control software to make a soft landing at sea. There is no way that it would be possible to access the exact software program that they use. It is probably extremely complex.

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