How does NASA know the exact position of James Webb Space Telescope at a given time?
Now that it has launched, NASA doesn't know the exact position of the JWST at any point in time. The three sigma position and velocity errors at MCC-2 were estimated to about 29 km and 21 cm/s, respectively. That's not exactly "exact".
Since they are now up to MCC2, how do they determine the exact time and force required to enter the L2 halo orbit?
Fortunately, NASA doesn't need to know the exact time and force. The project would be toast if they did. (Any project that needs "exact" position and velocity would be toast.) What NASA doesn't want to is to push the spacecraft so hard that a subsequent correction burn would require the vehicle to rotate to such an extent that the cold side becomes sun-facing. On the other hand, a slight undershoot is okay. I suspect the JWST Flight Dynamics Team will err on the side of caution and target a slight undershoot. (Or probably did err on the side of caution side as MCC-2 has now been performed.)
Is it a matter of good old fashioned dead reckoning?
Absolutely not. Relying on dead reckoning and nothing else is a good way of ensuring that the vehicle will soon be dead. I do not see any references that indicate that the JWST uses dead reckoning at all.
I see zero evidence that the JWST self-navigates its translational state. Instead what I see are papers going back to 2003 trying to determine whether the JWST needs accelerometers at all. If the spacecraft does have accelerometers, it would only be for the purpose of determining when to terminate a delta V maneuver. In particular, accelerometers would not be needed if the JWST Secondary Combustion Augmented Thrusters (SCAT thrusters) are so precise that the error between predicted and actual delta V is within 1.5%, but absolutely would be needed if the error is over 5% (three sigma).
I've yet to see a thruster whose predicted - actual delta V is less than 1.5%. Ive seen many where predicted - actual delta V exceeds 5%. My guess is that the JWST does have accelerometers, but that's just a guess. The literature on the onboard sensors JWST related to propulsion is rather scant. There's lots of literature on attitude and attitude rate sensors, but almost none on sensors related to propulsion.
In any case, I see no literature crowing about the JWST's onboard translational navigation capabilities. If the JWST was using dead reckoning, augmented of course by occasional corrections from NASA, there would be paper upon paper upon paper crowing about this new capability.
Stealing a term from software, YAGNI (You Aren't Gonna Need It) and KISS (Keep It Short and Simple) apply in droves to capabilities on a spacecraft. Most spacecraft do not know where they are because they don't need to do so and because Guidance, Navigation, and Control software oftentimes is the most complex and expensive software on vehicles that do need it.
The Space Shuttle and International Space Station software are extreme examples. Counting all of the people who wrote flight software, wrote unit tests of that flight software, wrote simulation code to further test that flight software, people who maintained the test environment, testers who ran the tests, and evaluators who poked and prodded at everything, the Shuttle and ISS flight software was written at the mind numbing pace of one line of code per person per week. I suspect the JWST flight software was written at the more typical rate of one line of code per person per day, or maybe even per hour. One line of code per person per hour would be phenomenal.
Even at this phenomenal rate of one line of code per person per hour, YAGNI and KISS would dictate that eliminating the many thousands of lines of mathematically complex, error prone, and computationally expensive flight software code needed for a spacecraft to self-navigate via dead reckoning is a good idea if that capability is not needed. And it's not needed in the case of the JWST.
What the JWST does need to do, with very high precision, with very high accuracy and with very high smoothness, is to know where to point itself and to point its telescope with respect to the "fixed" stars. There is lots and lots of literature on this JWST capability. The JWST team crows about these capabilities, and that crowing is very well justified.
What the JWST does not need to do is to know where it is. Knowing where the JWST is is a collaboration between NASA's Deep Space Network (DSN), managed by the Jet Propulsion Laboratory, and the JWST Flight Dynamics Team, managed by and hosted at the Goddard Space Flight Center. The DSN measures range (distance to the spacecraft) and range rate (time derivative of range) extremely precisely. Additional precise measurements can be made when multiple DSN ground stations can communicate with the spacecraft simultaneously.
The DSN provides these precise measurements to the JWST Flight Dynamics Team, which is responsible for determining the JWST state (position and velocity) and for planning mid-course corrections, orbit maintenance maneuvers, and momentum dumping maneuvers. This is the antithesis of dead reckoning. Techniques for precision orbit determination go back to Gauss, and the techniques have improved markedly since Gauss's time. Gauss's techniques involved azimuth and elevation measurements only; range and range rate were not available. Range and range rate are so precise for spacecraft beyond low Earth orbit that modern precision orbit determination techniques oftentimes ignore azimuth and elevation measurements as overly noisy parameters.
Anne Long et al., "Navigation concepts for the James Webb space telescope," 2003 Flight Mechanics Symposium (2003)
Sungpil Yoon et al., "James Webb Space Telescope Orbit Determination Analysis," (2014).
J. Levi et al., "The JWST Flight Dynamics Operations Concept and Flight Dynamics Ground System," 2020 IEEE Aerospace Conference (2020).