In this case, "robotic" and "private" could be similar enough in category to be confusing. In the Order of Adjectives[0], "robotic" is in the TYPE category, near the bottom of the list, and "private" seems to fit in that same category at first glance. By that interpretation, either "robotic private" or "private robotic" works.
What if instead of "private" it said "Californian"? That would make it an ORIGIN, and "Californian robotic spacecraft" becomes the obvious choice — otherwise, you'd think they were talking about a spacecraft belonging to robots from California. ;)
So if we interpret "private" as an ORIGIN, your "private robotic spacecraft" sounds better. That would have been my choice as well.
[0]: https://dictionary.cambridge.org/grammar/british-grammar/adj...