But if we examine it closely, what do we see? Hopefully answers will address the following:
- Does the command contain a set point for engine cut-off based on integrating one accelerometer, or three, or something else?
- Does the command include a direction in an inertial frame, or the spacecraft's frame?
- How would this technique be compared and contrasted to the famous "burn scene" in the movie Apollo 13?
The answers to the first two questions are "it depends". It depends on a lot of things, and it varies from spacecraft to spacecraft. The answer to the third question is that Apollo 13 shut down its flight computers and its sensors to conserve power. The better alternative of a commanded ΔV was not available, so they had to revert to a timed burn as the only alternative.
Since "it depends" is a key part of the answer, it helps to look at the question raised in the title: How exactly do “commanded delta-v burns” work in practice? (e.g. OSIRIS-REx)
The closest reference I could find on OSIRIS-REx is the eoPortal page on the vehicle, which says regarding a trajectory correction maneuver that
This trajectory correction maneuver was the first to use the spacecraft’s ACS (Attitude Control System) thrusters in a turn-burn-turn sequence. In this type of sequence, OSIRIS-REx’s momentum wheels turn the spacecraft to point the ACS thrusters toward the desired direction for the burn, and the thrusters fire. After the burn, the momentum wheels turn the spacecraft back to its previous orientation. The total thrust is monitored by an on-board accelerometer that will stop the maneuver once the desired thrust is achieved.
Some parts of this are incorrect. Accelerometers are somewhat dumb sensors. They sense non-gravitational acceleration. While they do accumulate sensed acceleration over short periods of time so as to smooth out randomness, they do not accumulate delta V over long periods of time. A small fraction of a second counts as a "long" period of time with regard to accelerometers. Moreover, accelerometers do not command thrusters. It is the spacecraft's on-board guidance, navigation, and control software that starts the burn, that monitors accelerometer output, and that stops the burn when the accumulated delta V reaches the commanded delta V.
From that brief and somewhat incorrect description, I suspect that what OSIRIS-REx does is that prior to a burn it transitions from barbecue attitude mode to inertial hold attitude mode. At the commanded time, it sets a variable typically called dvtogo (or some variant thereof) with the commanded ΔV and initiates a burn in a fixed inertial direction. At each time step, the GN&C software decrements dvtogo with either $(\vec a \cdot \hat d) \Delta t$, where $\hat d$ is the unit vector in the desired direction, or with $||\vec a||\Delta t$. Which approach is taken is not clear from the quoted paragraph; different spacecraft use different approaches. Either way, once dvtogo becomes close to zero or negative, the GN&C flight software commands the thruster to shut down.
The above approach is not quite optimal. A more optimal approach with regard to fuel consumption is to change the direction as the burn progresses so as to reduce gravity losses. This added complexity ("perfect is the enemy of good enough") is probably not worthwhile for a mission such as OSIRIS-REx. For a moon lander, the added complexity of approximating a gravity turn may well be worthwhile.
