Q. Is there a linkage in the plane that traces out a circle $C$ in such a manner that the interior of the disk bounded by $C$ is never intersected by any link througout the motion?

What I mean by "a linkage in the plane" is best illustrated by the famous Peaucellier linkage, which traces out a straight line segment: _{(Wikipedia image.)}
         
Some vertices are "pinned to the plane." All vertex joints are universal in the sense of allowing full $360^\circ$ motion. All links are rigid segments, which can pass over one another in a physical, layered model, e.g.,
         
          _{(Image from this How Round Is Your Circle web page.)}

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Of course it is trivial without the restriction that the links not intersect the interior of $C$: One radial link pinned to the center of $C$ suffices. And the challenge in the Peaucellier linkage was to convert the natural circular motions of linkage components into straight-line motion. Here I am seeking a vertex of the linkage to follow a natural circular motion, but with the restriction to not intersect the interior of that circle $C$. Following a subarc of $C$—say, a semicircle—without intersecting the interior of $C$, would also be quite interesting.

It may be that the 19th-century masters (Peaucellier, Lipkin, Watt, Chebyshev, et al.) did not investigate this question. But perhaps there is an easy construction I am not seeing...?