I'm mostly just going to do some tweaked slice-and-dice here from my new book. It allows me to self-promote shamelessly and it also saves time, considering that I'm an agonizingly poor typist.
The intent with this skill is to encourage the unrolling loop to loft from a low, side-arm position. Once the loop unrolls, the line (particularly the end of the line) drops in small curves. The application: Because the line stays low for the majority of the cast, the Pendulum is a slick way to get useful slack under overhanging branches, bridges, etc. It can also be used to drop a heavier fly more softly (flats anglers pay attention to that one), and as a general replacement for a Side-Arm Puddle Mend or similar skill.
The Pendulum Cast is made in a side-arm position (often with a palm-forward orientation) with a stroke that generates an underloop. Viewed from the casting arm side, the forward stroke/cast tracks through a shallow arc rather than a straight line.
This can be like a stretched crescent, or it can be more of a deeper ) shape lying on its side. In addition to needing some rod lift to get going (if you want a Pendulum backcast to start), this stroke path requires that the the wrist also follow along, and not be directed straight forward. This has some similarities to an Up-Hook Curve, if you're familiar with it.
Make sure that you don’t swing the Pendulum too low, either, or your underloop will tag the water’s surface. You may need to raise your entire casting arm somewhat to avoid this problem.
The cast is finished by proactively following the line to the water with the rod.
Since the Pendulum is built within the casting stroke, it can be considered a type of Puddle or Pile Cast (versus a Puddle or Pile Mend, which has a different basis).