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Consider the space shuttle or the Saturn V.

Launch would begin with an open-loop pitch schedule obtained from simulation with the day’s winds.

The pitch schedule would be specified as a lookup table with time-pitch breakpoints.

How far apart on the time axis would those breakpoints be? If the navigation-guidance system ran at 25 Hz (period of 0.04 s), then would the pitch schedule breakpoints be specified at 0.04 s intervals?

If yes, then maybe you wouldn’t need to interpolate between them, since the table would already be at the finest resolution the controller could follow (the 25 Hz).

But if no, then you would need to interpolate those breakpoints. And the interpolation could be linear, cubic, etc.

I’m curious which interpolation method the space shuttle/Saturn V would have used? Linear is simplest and those rockets weren’t rich in computing power... but could it have been a cubic spline interpolation method? Some other type?

If you know and can share—-thank you :)

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    $\begingroup$ My previous comment was wrong. Looking for a source now... $\endgroup$
    – Organic Marble
    Commented May 29, 2021 at 3:42
  • $\begingroup$ "time-pitch breakpoints". Are you sure that time is the independent variable ? It could be velocity or dynamic pressure based. Why time ? $\endgroup$
    – AJN
    Commented May 29, 2021 at 4:25
  • $\begingroup$ 0.04 s is quite small. I would think that the time difference could be as large as 0.5s or even 1s. There day's wind measurement will have quite a lot of uncertainty in it. The base wind data won't have high temporal (or spatial) resolution to justify storing the data at 0.04s. $\endgroup$
    – AJN
    Commented May 29, 2021 at 4:28
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    $\begingroup$ Slerp. Apollo didn't use slerp as the technique was invented in 1998, and I suspect the Shuttle didn't either because it took a while for the concept to make its way out of the computer graphics community to aerospace. The concept did make its way out of the computer graphics community to aerospace; modern spacecraft oftentimes use slerp. $\endgroup$
    – David Hammen
    Commented May 29, 2021 at 7:06
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    $\begingroup$ And yeah, @DavidHammen. I know! Lots has changed since the Apollo days and even since the space shuttle days. But. My hands are tied. You know I'll never get what I need if I ask about Falcon 9 or New Sheppard or Electron's/ULA's rockets. They are all super tight-lipped. So I look for what I know I can find. plus: the space shuttle and apollo and saturn v papers are so well written and complete and understandable. Most everything else on the topic is cryptic and detached from practical simulation. I've learned tons and tons from the shuttle/saturn v/apollo, so I know they're good sources. $\endgroup$
    – user39728
    Commented May 29, 2021 at 7:25

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I couldn't find a public source so consider this a "hearsay" answer

  • The shuttle first stage maneuver table consisted of 30 points
  • The independent variable was velocity, not time.1
  • Simple linear interpolation was used.

Source: Worked on DOLILU for two shuttle missions

1This is confirmed in the Ascent Guidance Training Manual paragraph 3.2.1

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    $\begingroup$ A number of modern launch vehicles still use this or a similar approach for first stage open loop guidance. Closed loop guidance (e.g., Powered Explicit Guidance) starts after MECO or stage separation. $\endgroup$
    – David Hammen
    Commented May 29, 2021 at 16:43
  • $\begingroup$ Thank you! This is so helpful. 30 breakpoints isn’t huge. This would be from launch through MECO? How long would that be on a typical shuttle launch to the ISS? I want to guess 30 points over maybe 120 s for 4s intervals? Also: linear interpolation, but passing through a low-pass filter which would round out the kinks, right? But would that be enough for 4s linear segments? I can’t help but imagine that would look staccato-like, unless you the smoothing low-pass filters had a large time constant, but even then... would the filter eliminate the derivative discontinuity at the breakpoints? $\endgroup$
    – user39728
    Commented May 29, 2021 at 23:26
  • $\begingroup$ Huge thanks for confirming the independent variable was velocity. Can you say if I’d that velocity was inertial (relative to ECI frame), ground velocity (relative tobECEF frame), or maybe “air velocity” (relative to the local winds)? I want to guess inertial velocity but... I’m wrong a lot. $\endgroup$
    – user39728
    Commented May 29, 2021 at 23:30
  • $\begingroup$ 1) the 30 points were for 1st stage (until SRB sep) - after that it was PEG guidance. 2) It was "ground relative velocity", IIRC it means velocity in the Boost Reference Frame. $\endgroup$
    – Organic Marble
    Commented May 30, 2021 at 0:00
  • $\begingroup$ Boost Reference Frame was an inertial frame with the Z axis through the launch site at liftoff time , X axis pointing north, Y axis pointing east. The 30 yaw, pitch, roll angles were in this frame. $\endgroup$
    – Organic Marble
    Commented May 30, 2021 at 0:08
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The Launch Vehicle Digital Computer (LVDC) in the Saturn V used quartic polynomials, four segments of them, for the pitch program, with the angle being a function of time since liftoff.

Saturn V Pitch Program

Taken from Saturn V Launch Vehicle Guidance Equations page 4-14. F are the coefficients of the polynomials, $t_c$ is the clock time since liftoff and $\Delta t_f$ is a term used to freeze the pitch program in case of an engine failure.

An example set of coefficients for Apollo 11 can be found in the AS-506 Launch Vehicle Operational Trajectory page 4-10, here the first of four segments:

enter image description here

The polynomial is evaluated every major loop, which contains the majority of calculations to be done by the Launch Vehicle Digital Computer. The major loop execution time isn't fixed, but takes about a second to run. The actual steering (output as an attitude error to the analog Flight Control Computer) is calculated on the fixed cycle time that you already mentioned, the 25 Hz.

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  • $\begingroup$ This is so interesting! First off, that Saturn manual is gold. I’m dying to sit down and just skim through it. I saw it covers logic in the TOC! I have an open question on that exact topic. But for this question, I’m happy to see they used time as independent variable, because it suggests my first guess wasn’t so unreasonable after all. Also very interesting interpolation algorithm, and one that would have been continuous across the breakpoints, it seems, since it’s fourth order? That would seem a boon. Anyway, this is all excellent to know. Thank you for posting!! $\endgroup$
    – user39728
    Commented May 29, 2021 at 23:34

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