Michael descends from Matthew Clarkson, provincial secretary to the colony of New York, in a long line to his great-grandmother Vida Hamlin (Clarkson) Ridpath. Matthew Clarkson has a known royal line from David I, King of Scotland (d. 1153), first published in the New York Genealogical and Biographical Record 127 (1996):193-201. "Improvements" to a royal line merely means that more research has been done and you can connect your gateway ancestor to a monarch who lived less long ago.
The first improvement for Matthew Clarkson shows that his Baron Grey of Ruthyn ancestors have a Longespee descent from Henry II, King of England (d. 1189). This appears in the latest edition of Gary Boyd Roberts's RD600, the addendum for 2008.
The second and ongoing improvement is being discussed now at SGM (soc.genealogy.medieval) and may lead to a descent from Edward I, King of England (d. 1307) through the Kendricks. It a good posting with sources.