I have never seen the original Iron Byron machine. I have only seen one version of that type of club-testing machine at our local golf practice facility. It is a very advanced machine, which they call Robo-Fit. It has an universal joint that allows the clubshaft to rotate about its axis.
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I don't have a conclusive answer as to how the clubhead swivels automatically in an Iron Byron-type machine that has an universal joint. I have previously thought that it relates to the COG of the clubhead's mass relative to the hosel, and I have presumed that the offset COG of the clubhead causes the clubhead to swivel around the clubshaft's axis. If you establish a conclusive explanation, please let us know.
Here is a link to the company that makes the machine.