I'm not sure others have read deeply into the article. While there's always some subtle ways friction can sneak its way, the authors account for the obvious.
The article has a lot of definitions, which makes the crux difficult to pinpoint. They use the word "shape" many times to describe the configuration of the four motors. I think this adds confusion. Let me try to summarize as I understand it without using the word "shape."
The authors are measuring the net position of the arm holding the two motor tracks. The arm can only rotate clockwise or counter-clockwise in the horizontal plane, so it suffices to measure the arm position using an angle - phi, with initial position being phi = 0.
The two motor tracks are oriented perpendicular to one another. A key point is that the motors do not travel up, down, left, or right. That's Cartesian thinking. Rather, they travel clockwise or counter-clockwise along their respective tracks. One track is vertical, the other is horizontal.
With this picture, the authors claim that the net horizontal position angle (phi) of the arm changes depending on how the motors move along their tracks.
None of this is particularly impressive (e.g. the motion of the motors may cause the arm to "wobble" one way due to issues with synchronization, weight differences, and numerous other causes).
The interesting part is this: If the vertically-oriented motors are made to move up-down on linear straight paths instead of curved paths, the net movement of the arm is not observed. Think about that for a moment...why should the path that the vertically-oriented motors take affect the horizontal rotation of the arm?
In Newtonian physics, acceleration/force in one direction cannot affect acceleration/force in a perpendicular direction. At any given moment, the vertically-oriented motors are perpendicular to the arm's rotation direction, so their motion should not affect its horizontal rotation in space. In particular, both the curved-vertical and linear-vertical paths are perpendicular to the horizontal at all times. Newton would predict that changing the path of the vertical motors from linear to curved would not affect the horizontal rotation...but it does.