Ok, I can finally re-read this now without my head shaking so violently that it prevented me from typing.
Patience in the face of abject incomprehension is a virtue. Not patience
with it, but in the face of it.
The Saturn V used a space-fixed IMU for its INS.
Indeed, and the accelerometers at the center of it need to be aligned in a particular way at the instant of launch. Why? Because of the way we evolved our expertise in launch vehicle control. A launch vehicle (whether for an ICBM or a commercial vehicle headed for space) wants to navigate according to control laws that borrow a fair amount from fixed-wing airplane control.
The classic airplane motions -- roll, pitch, and yaw -- form the basis of launch-vehicle guidance on ascent. Consider the vehicle two minutes or so into the flight. To simplify, you roll the vehicle so that its notion of up (despite its cylindrical shape) is the local "up" reckoned by the horizon. In that orientation, pitch determines the velocity and altitude profile. Yaw determines conformance to the desired ground track.
Each of those rotations is reckoned according to its own independent set of control laws. This is not the same way a spacecraft in free space is oriented and controlled. A rocket is first an airplane and must be flown somewhat as such. But in order to make those control laws as robust as possible, and their input values as pure and free from error as such, the accelerometers are physically oriented so as to describe an orthogonal reference frame at the instant and precise location of launch.
To simplify: Positive X is up -- that is, the local up at the launch site at the time of launch. It points straight out from the Earth parallel to the rocket. Positive Z is the launch azimuth. (More on this later.) The ascent path lies in the X-Z plane along with important conceptual quanties such as the thrust vector, the drag vector, the diminishing gravity vector, the angle of attack. Which is to say, we want to consider them as much as possible as 2D quantities with 2D solutions. We want the X and Z accelerometers alone to tell us about lift and downrange acceleration without gimbal lock. We want to be able to integrate just those values, without quaternions or basis changes or other complications. Control laws work best when they are dirt simple.
The Y axis, predictably, is perpendicular to the ascent plane and represents left and right deflections from the desired course -- such as from vehicle elasticity, thrust irregularity, wind, etc. The Y axis conceives of the flight path as seen from above, without respect to altitude or speed. There is only left or right.
So we have roll control, yaw control, and pitch control. With the platform properly aligned at launch, and certain controllers properly functioning, we have dedicated accelerometers whose sensitive axes correspond directly to the inputs to the control laws.
The roll control law is basically "Keep the heads-up axis of the rocket pointing directly away from Earth's center." The heads-up axis is a direction perpendicular to the longitudinal axis of the vehicle, chosen arbitrarily and then forming the basis for other sensor and structure location. Some launch vehicles use optical horizon sensors. Some use dedicated roll gyros. The point is that roll control is a simple control. Error and error rates in this channel translate to thrust vector commands to treat the outboard engines as if they were airplane ailerons. Roll control doesn't give a [bleep] about altitude or whether the rocket is on course.
The yaw control law is basically "If we drift left, steer right; if we drift right, steer left." Again, dirt simple. The Y accelerometer alone drives this process in the ideal world. The output of the control law are commands to treat the steerable outboard engines just like and outboard motor on a boat.
With roll and yaw control achieved by those simple control laws, pitch control can remain a 2D solution. And the pitch control law is a bit more complicated. When the launch vehicle engine shuts off, the precise geometrical point at which that occurs, and the velocity vector that prevails at that instant, together define the orbit. Hence the goal of the launch vehicle is to get to a certain altitude, a certain distance downrange, and a certain horizontal and vertical velocity -- all in the conceptual X-Z plane, but realistically speaking in Earth-fixed 3D coordinates.
So the pitch control law has to make sure the rocket climbs appropriately according to the ascent profile. Since it has to trade speed for altitude, that's an important control law to get right. That's where the majority of expertise is applied, and the majority of gentleness and finesse such as easing the AOT around max Q.
We launch rockets
roughly eastward. The exact azimuth depends on the desired orbital inclination at insertion. That in turn varies by the launch instant within the launch window. As the window opens, the launch azimuth is set at a particular value, and changes slowly as the window progresses. This is why the Z axis has to be updated at the very last second; you don't know for sure what the launch azimuth will be.
In practice the first few seconds of flight don't follow the general guidance rules. Until the tower is clear, the vehicle generally flies straight up, or according to a programmed tower-avoidance manuever. Then it rolls into the departure azimuth. For example, the space shuttle sat on the pad with the vertical stabilizer pointing south. It rolls to point the vertical stabilizer down the launch azimuth. This begins the formal closed-loop ascent guidance.
Indeed the notion that the IMU needs to be frequently updated during the ascent is hogwash. It receives updates prior to launch only to ensure that the X-Y-Z axes correspond as exactly as possible to the launch site and ascent profile orientation at the instant of launch. Ballistic missiles operate the same way, only with more generalized notions of launch azimuth. They still align the platform shortly before launch and then don't need to again until (in second-generation boosters) the opportunity for a star-sight refinement occurs.
