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This answer to the question What was the mascon “fix” used by Apollo 12? says (in part):

The Tindallgrams have some notes on the matter. See pages 307 to 354 of http://www.collectspace.com/resources/tindallgrams/tindallgrams02.pdf

Briefly, improvements were made in a couple of areas:

  • The model of the moon's gravity field (based on tracking data from earlier Apollo and Lunar Orbiter flights)
  • Real-time estimation of the spacecraft's trajectory - a new "Lear Processor" was brought online at Misson Control's Real-Time Computer Complex, which was running a specialized Kalman Filter to provide optimal use of all the tracking information. (emphasis added)

The linked PDF (quite an interesting read if you have the time!) does contain more than a dozen instances of "Lear Processor" peppered throughout it's almost 400 pages, but I can't find a clear explanation of what it is. It might be related to the aerospace company Siegler or Lear Siegler (merged in 1962), but I don't know for sure.

Question: Is it possible to explain what the "Lear Processor" is? Is it software, hardware, or both? Is it an early implementation of a Kalman filter only, or does it contain/do a lot more?

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It's an algorithm to speed up certain matrix operations, not a piece of hardware. It appears that the algorithm was first invented by William M. Lear for the LM guidance computer, as explained in @Uwe's answer. However, the programmers at the Real-Time Computing Center back in Houston then adopted the algorithm to process tracking data of the LM, naming it the "Lear Processor" after its inventor. The algorithm was used to speed up a Kalman filter and may have had other uses.

Introduction to trajectory estimation for RTCC programmers is a training paper for programmers in the RTCC. On the last page is bibliography entry #12:

  1. Lear, W.M., deVezin, H.G. Jr., Wylie, A.D., and Schiesser, E.R., RTCC Requirements for Mission G: MSFN Tracking Data Processor for Powered Flight Lunar Ascent/Descent Navigation, MSC Internal Note No. 69-FM-36 (Feb. 7, 1969). Houston, Texas: Manned Spacecraft Center.

Lear's paper (thanks @indy91) is cited several times in the paper that I have linked above. Section 17 (MODIFICATION OF THE STATE COVARIANCE MATRIX) says

All we have done is add a constant to a main diagonal element. This is the method used in the LM powered flight processor [12].

It's also cited at the end of section 20 (EXPONENTIAL DOWNWEIGHTING OF PAST DATA):

The Bayes filter is particularly well adapted to estimating free-flight trajectories of long duration, where the observations actually are received in batches. Then each batch can be processed as it is received to update the state vector. The Kalman filter is particularly desirable when the observations are coming in continually and the trajectory characteristics are such that point-by- point processing of data is required, e.g., the LM powered flight processor. [12]

The Bayes filter requires inversion of matrices with order of the state vector; the Kalman, with order of the measurement vector. So the Kalman is very useful in avoiding inversion of large order matrices. For example, in the Kalman filter, LM, powered flight processor [12] the state vector has 21 elements; the measurement vector, 4 elements.

Section 22 (CORRELATED DOPPLER MEASUREMENTS) says

This way of processing the pseudomeasurements was presented to show how it can be done, but it is really clumsy compared to the following equivalent method which uses the actual measurements [12].

If you really want know the mathematical details, they're in the paper I have linked above.

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According to this page William Lear, an aerospace engineer wrote a Kalman filter program for Apollo 11 Lunar Module computer.

In particular, the on-board computer that guided the descent of the Apollo 11 lunar module to the moon had a Kalman filter. That computer was also communicating with a system of four Doppler radar stations on Earth that were monitoring the module's position. It was important that estimates from all sources be good: The Earth-based estimates were used to adjust the on-board system; if they had disagreed too much with the on-board estimates, the mission would have had to be aborted.

According to William Lear, an aerospace engineer who was then at TRW in Redondo Beach, California, NASA contacted him about nine months before Apollo 11's scheduled launch because their Earth-based tracking program wasn't working. Lear, who now works for Draper Labs at the Johnson Space Center, wrote a 21-state Kalman filter program, which went into the Doppler radar system. The final check of the program, Lear recalls, was done the day before Armstrong, Aldrin, and Collins took off.

See also KALMAN_FILTER.agc and Navigation Filter Best Practices Bibliography

[39]W. M. Lear. Multi-phase navigation program for the space shuttle orbiter. Internal Note No. 73-FM-132,NASA Johnson Space Center, Houston, TX, 1973.

[40]William M. Lear. Kalman Filtering Techniques. Technical Report JSC-20688, NASA Johnson SpaceCenter, Houston, TX, 1985.

From the oral history project of EMIL R.SCHIESSER:

Stan Schmidt was working with us and implemented equations to account for imperfect forward propagation of position and velocity due to imperfect knowledge of the forces acting on the vehicle to create a viable Kalman filter for navigation purposes. Around 1964 Bill [William M.] Lear from TRW started to help us. He worked, I think, in Redondo Beach, California.

I asked him to work on the development of Kalman filters for the various Apollo navigation tasks. He was a really smart guy and easy-going; smoked a pipe, professor type, Dr. William Murphy Lear. From then on and throughout all of the other programs he was the one we relied on for all our Kalman filter formulation and design. He could do more work in two months than a team of five people could do in six, and it would be better. This might be a bit of an exaggeration. But then he tended to work day and night.

Bill contributed several advancements to our Kalman filter design, including:
'measurement underweighting' to account for the use of linear assumptions for a non-linear relationship between deviations in a measurement to deviations in the local position and velocity; forcing the estimated uncertainty matrix to be symmetric; and the use of exponentially correlated random variables for modeling state parameters related to systematic error. The Kalman filters developed by Bill are sometimes referred to as Lear Filters or sometimes the Lear filter, though there were more than one.

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    $\begingroup$ "The final check... was done the day before Armstrong, Aldrin, and Collins took off."!!! $\endgroup$ – uhoh Jul 8 at 11:19
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    $\begingroup$ I'm sure this is the right Lear, but the question asks about a program or processor in the MCC, and this answer talks about an onboard program or processor. Is one just a copy of the other? $\endgroup$ – Organic Marble Jul 8 at 12:46
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    $\begingroup$ More information about the right Lear is hard to find, I got many false hits about 'King Lear' by Shakespeare. $\endgroup$ – Uwe Jul 8 at 14:25

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