Ellipse                                                                                                      Latest change 2023-12-25

In brief:
One of the biggest problems with a Foucault pendulum is that it tends to follow an elliptical path, even when it was launched in a perfect straight line and there is a perfect rotational symmetry in the whole construction. This ellipse has it's own precession which can easily overwhelm the Foucault precession.
This problem is bigger for shorter pendulums. There are several methods to suppress ellipse forming c.q. the precession of it.

Also look in Ellipse Thoughts
Kamerlingh Onnes describes in his thesis a.o. that the ellipse will periodicaly change direction. 

The so called Charron Ring is a ring, placed somewhat below the top, which the cable of the pendulum touches slightly with each swing. This limits the amplitude of the swing, but also -and stronger- supresses ellipse forming, because at the moment of touching the velocity of the unwanted ellipsoical component is at its maximum and so is reduced the most by the friction.
The drawback of this method is that the ring sits in a very high location, often difficult to access and it must be adjusted quite precise to prevent a preference for the direction of the swinging plane. A second drawback is that the mechanical contact introduces wear on the cable with the risk of breaking.
For the pendulum in the UN-building in New York this principle is also used, but with a totally different implementation which eleminates wear almost completely.
I also heard about variants where the ring is covered with a rubber like lining. That may damp the ellipse perhaps better and eliminates wear, but these materials themselves may become brittle or pulverized. Not a good idea if you cannot access it.
This company uses on the cable near the top a cylinder with 2 O-rings which, according to the installation manual, must make a solid contact with a fixed safety collar. O-rings probably have an acceptable long liftime.

Another effect of the Charron ring is that it reduces the average period time of the long axis of the ellipse because during part of the swing the effective cable length is reduced. By adjusting the amount of energy delivered by the drive coil it might be possible to make the long and short-axis swing times exactly equal, thus eliminating the precession of the remaining ellipse.

Salva and some others place a thick metal ring (preferably copper) such that a small magnet in the bob flies over it at the maximum amplitude. The eddy-currents in the ring will damp the movement somewhat, but again the damping is stronger for the elliptical component.

Szostak states that a repelling drive will lessen the ellipse and an attracting drive will amplify it. He does not provide arguments for this except a little drawing which t.m.h.o. shows the opposite. Several sources I found have conflicting opinions on this.

Witzel  puts a weakly attracting magnet below the center. For the movement in the long axis this has no effect, the bob is accellerated a bit when approaching and decellerated the same amount when leaving. But for the short axis of the ellipse the force is always attracting, unless the short axis becomes zero, and that's just what we want.
A variant could be to engage the drive coil in attracting sense, symmetrically around the center.

Crane also mentions a permanent magnet in the center, but this time repelling. This would not suppress the ellipse but, when correctly adjusted, the precession of the ellipse. Crane uses attracting and repelling drive (switch over at center passage) which he hopes should eliminate the effects of certain asymmetries in the system, which he does not mention.

Schumacher states that repelling drive counteracts the precession of the ellipse (not the ellipse itself) and he derives that activating the drive coil when the bob is at a certain distance d from the center, the precession will be suppressed completely.
The distance d depends only on the Length, the Amplitude and the Q-factor of the pendulum.
My experiments up to now tend to confirm this. In all pendulums I operated up to now I've seen alternating ellipses, changing direction around E-W and around N-S. When the impulse is given to early the ellipse precession is overcompensated, that is, during a CCW ellipse the precession goes CW and during a CW ellipse the precession goes CCW. When the kick is given to late it is just the opposite.

Some sources, like Pippard, suggest that a parametric drive, also called vertical or piston drive, would counteract ellipse forming. The trick is to lift the cable's mounting point a bit when the bob passes the center and let it go at maximal amplitude. Very much like driving a playgound swing by lifting or lowering your body's center of gravity.

Another method, related to an invention of Huygens, could perhaps also work. Here a small trumpet-shaped tube sits below the mounting point of the cable. As the pendulum swings out of the center, the effective length of the cable is gradually reduced an so the swing time decreases. With the proper shape of the trumpet the swingtime could perhaps become perfectly independent of the amplitude, and then the mechanism which transfers energy from the long to the short axis of the ellipse is eliminated and with that the precession of the ellipse. The art is to find -and machine- the proper shape of that trumpet.
(dec 2023) We know now that this shape is a part of a cycloid.
Look at some details of the pendulum in the Thij College in Oldenzaal (NL).