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1. Why in figure 27 the routine stores 3 velocity errors:
(Vel_err1,Vel_err2,Vel_err3)
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2. Why in the diff_err we have this form:
(vel-err-vel_err3+3(vel_err1-vel_err2))
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It appears that Vel_err, Vel_err1, Vel_err2, and Vel_err3, are a sliding sample window of the 4 most current velocity errors. I could't find any mention of their choice in the application note but it appears to the a heuristic to give the center sample more corrective action than the oldest and most current velocity error. That could be to damp-out sharp edges in the errors; something like computing convolutions with a geometric window.

I checked a few sites but none used the same formula in that area. Here some of the sites:

http://www.tech80.com/library/17.pdf
http://www.darnell.com/archive/pdf/aa_110899b.pdf
http://www.darnell.com/archive/aa_110899b1.stm

-Jay C. Box