Periodic error correction in automatic telescopes

Quoting the most relevant parts from the Richard McDonald's Wiki on Astrophotography Mounts: Periodic Error Correction:

Periodic Error can be reduced to an acceptable level using a variety
of techniques, only some of which are in the range of a beginner or
mid-level astrophotographer.

• Throw Money at the Problem

  Very high-end mounts for astrophotography have very small periodic error because of the time
  and money spent on manufacturing high-precision gears. You can also
  buy higher-precision gear upgrades for some mounts. For the beginner,
  let's call this impractical. Buying a $1000 replacement worm gear for
  your $1000 mount is probably not a good balance.




• Don't Use Gears

  This is really an extreme case of the "throw money" solution, mentioned just for fun. There are some
  very-high-end experimental drive systems showing up on the market that
  don't use worm gears, and that don't exhibit periodic error. Examples
  include direct drive systems and harmonic drives. 'Way out of our
  price range.




• Autoguiding

  A second guide camera and computer can be used to make frequent small corrections to the mount's pointing,
  and this can reduce or eliminate Periodic Error if the onset of the
  error is not too sudden. This is the subject of a separate article.




• Periodic Error Correction Feature

  Training PEC with hand-control Most mounts intended for astrophotography include a
  feature called Periodic Error Correction (PEC), which can be used
  alone, or in conjunction with Autoguiding. PEC is used in two phases:

  
  
  
  1. Training. During this phase you use the control panel or menus to say "Hey! Pay attention to this!" to the mount, then you manually
     keep a star perfectly centred for several worm periods. You do this by
     centering a star with a high-magnification eyepiece that includes a
     cross-hair reticle. You then stare at the star for 10 to 15 minutes
     and use the mount's hand controller to manually make the small
     adjustments necessary to keep it perfectly centred on the cross hair.
     The mount records the error corrections you supply, remembering where
     in the worm position each one was needed. By training through more
     than one worm cycle, the mount can record an average correction, in
     case you reacted slowly or over-reacted.
  
  
  2. Playback. Once you have trained the mount, you can turn on PEC. The mount will "play back" the recorded error correction information
     by slightly changing the speed of the Right Ascension drive to move
     ahead or back each time your manual error corrections did the same.
     This will cancel the periodic error, resulting in a smoother track.
  
  

^{Training PEC with
hand-control: Periodic Error Correction (PEC) button on the control
panel of an equatorial mount.}

The manual training phase is quite tedious and a more modern
alternative is to use a camera and computer - usually the same one you
will use for auto-guiding - to track a star while the mount records
the training information.

There is even specialized software available to help collect Periodic
Error data, analyze and smooth it, and upload it to the mount. You do
not need such software, as the manual techniques mentioned above will
work just fine. However, it makes a tedious job simple and pleasant,
and you may find it a worthwhile investment. I use PEMPRO and am
very impressed with it. It's not free, but it is inexpensive and works
very well (and includes another feature to help achieve perfect polar
alignment).

Quoted text and photograph source, copyright and courtesy of Richard McDonald, no copyright infringement intended. I would also highly recommend reading same author's extensive article on Autoguiding.