.... Ummmm, I sincerely doubt it.
Even with a SUPER precise position monitoring system in place (either a home-brewed error correcting ultrasonic/infrared hybrid comes to mind, or
you'd maybe want to purchase an off the shelf LIDAR system(s)) AND some how interface it with high end racing flight control equipment using an
ultrafast FC with some ridiculous loop time AND the latest ESCs AND an advanced feedback system in place (which, btw, does not exist in normal
quad set-ups other than the ubiquitous PIDs on an F4 or F7 scenario) you will never be able to maintain flight stabilities on ANY axis that will hold true
+/- one sixteenth of an inch !
The best attempt(s) would require either incorporating an auxiliary MCU hacked into the quad's normal (high end) flight control mechanisms via custom programming, backed with some predictive positional computational algorithms, along with custom hardware, or basically building a superior FC system from scratch.
Notice that even in the impressive demonstration given that quad did NOT have the control resolution you're asking for.
Some more information on your requirements regarding this project would be helpful in determining it's viability, e.g. :
a.) How many precision axis ? (vertical only, or more)
b.) Why 1/16th inch is critical.
c.) Your electronics expertise
d.) Your computer programming expertise.
e.) Your Calculus expertise.
f.) Your budget.
