This does in fact simplify the problem greatly. There is an effect that kicks in when the airframe is rotating that effectively provides a phase shift in where the control needs to be applied. In helicopters this can be up to around a 90 degree shift which really complicates the pitch/yaw control (think that controlling the blade pitch when abeam actually works on the pitch axis). Stopping the roll first eliminates that problem. In reality you can just start all controls up at once and the lower inertia about the roll axis means it stabilises quite quickly unless your control system is really badly wrong.
I agree that being able to correct for the rotation around the main axis does simplify the problem greatly.
But the problem with doing this is that the axial rotation isn't present to be measurable right off the rail. The forces causing rotation are somewhat in proportion to the rocket's velocity and with air resistance.
In contrast, the rocket tilt deflection angle from vertical happens right off the rail when the rocket hits a sideways air current and starts to "weather-vane". The rocket is more prone to "weather-vane" at slower velocities, and the tilt angle accumulates over time. Tilt control, like rotation control is better with higher velocity air flowing over the fins.
Although controlling the rotation first is desirable, it will need to happen throughout the flight unless the initial control of it that fixes stops the rotation completely. And a rotation fix would also require an integral component to integrates the rotational error measurement over time, which is time when the rockets tilt error also needs to be controlled. There's a small window of time when both rotation and tilt need to be controlled. In addition there's also gravity-cause tilt component vector that happens with long burn engines, and is the effect of gravity on the rocket's Cg.
But I do not underestimate the complexity of the multivariable control, especially with the need to determine the precise rotational orientation of the rocket tilt to determine which fins to adjust. While I've designed and tuned dozens of control systems with multiple single control loops and seen numerous analyses for multivariable control in advanced process control text books, I have only done one or two actual multivariable control systems throughout my career. In practice this is very difficult to do robustly.
One thing that might simplify the actual design of such a control system (that doesn't have separate modes of control for tilt and rotation) -- is that rotation might be both more easy to measure and to control throughout the flight. On the other hand, controlling the tilt angle is inherently more difficult to measure and control. The absolute tilt angle with reference to the vertical angle to ground must be deconvoluted from the rotational angle of the rocket's principle axis at every point in time. Then the correction to the tilt angle must be determined by adjusting the relevant fins. A four-fin model would simplify this over a three-fin model, but there's always a time phase angle both in the measurement delay and the time it takes to adjust the fins.
So by default, the rocket tilt control might have an indeterminant solution, and a large time phase shift would make the solution unworkably unstable, and the tilt control would be locked-out until the rotational spin is slow and under control. What this means is that it might not be necessary to have a modal control system decide to do rotational control first to simplify things. The result of having too fast a rotational spin would be that the tilt control is automatically prevented until a reliable tilt measurement is made and that the fin control for it can be safely accomplished.
Another relevant issue that I haven't addressed with this "PID Tuning Help Thread" -- is what are the instruments actually being used to measure the rocket's rotation around it principle axis, as well as for measuring the tilt angle off of vertical? Do instruments actually exist that are affordable to rocketeers that provide data with the resolution and precision to control from? A while ago I saw a demonstration video of some kind of control system that controlled both rotation and vertical tilt but haven't been able to locate it since.
The value to be controlled is the rocket's tilt angle deflection from the vertical, rather than pitch or yaw. But what instrument available to rocket hobbyists can actually measure this to the extent that it can be the basis for control? Its there an affordable "gyroscope" chip with both the precision and the ruggedness to use in high power rocketry where the acceleration can be in excess of 20 G's?
To measure the axial rotation, I'm presuming that a 3-axis accelerometer chip similar to the one that Jolly Logic uses in their Altimeter3 could do the job. When aligned with the z-axis parallel to the rocket's longitudinal axis, the root sum square of the x and y axis acceleration is proportional to the rotation rate. The output of this chip could probably be scaled to the maximum rotation rate that's ever likely to occur.
It would be nice to have a discussion from others on this thread on what they know about measuring rocket tilt angle and rotation. You cannot control what you cannot precisely measure. I'd welcome learning that reliable tilt and rotation instrumentation exists to do the measurement and control, and that I am only badly misinformed about them.
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