Correcting Accelerometer Readings when Offset from Center of Mass: How?

Question

A follow-up to an earlier question.

So I've accepted that I need to correct my accelerometer readings for the accelerometer offset from the center of mass.

The offset means that whenever the rocket rotates, the accelerometer will read an acceleration, even though the center of mass is not accelerating---and it's the center-of-mass acceleration that I need from my accelerometer.

I can calculate the rotation terms and subtract them from my acceleration readings to estimate the acceleration of the center of mass, which means I have to calculate also the location of the center of mass as it shifts down due to fuel consumption.

But I wonder how the correction is done in practice. I'm looking for technical detail, an algorithm even, if it's public... but at least a solid technical description of the algorithm. I want to say this is buried in the many space shuttle and saturn v and apollo papers now in the public domain... but I haven't had the good luck of running into it just yet.

Any pointers on where I might find this info? Huge thanks if you can help!

Answer 1

This answer on the Physics sit seems to sum up the maths pretty well:

With the accelerometer A and the center of mass C we have $\vec{c} > = \vec{r}_C - \vec{r}_A$.

$$ \vec{a}_C = \vec{a}_A + \dot{\vec{\omega}} \times \vec{c} + > \vec{\omega} \times \vec{\omega} \times \vec{c} $$

one can you the 3×3 cross product operator to transform the above into

$$ \vec{a}_C = \vec{a}_A + \begin{vmatrix} 0 & -\dot{\omega}_z & > \dot{\omega}_y \\ \dot{\omega}_z & 0 & -\dot{\omega}_x \\ > -\dot{\omega}_y & \dot{\omega}_x & 0 \end{vmatrix} \vec{c} + \begin{vmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x > \\ -\omega_y & \omega_x & 0 \end{vmatrix} \begin{vmatrix} 0 & > -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{vmatrix} \vec{c} $$

or in the form seen the linked post

$$ \vec{a}_C = \vec{a}_A + \begin{vmatrix} > -\omega_y^2-\omega_z^2 & \omega_x \omega_y - \dot{\omega}_z & \omega_x \omega_z + \dot{\omega}_y \\ \omega_x \omega_y + \dot{\omega}_z & > -\omega_x^2-\omega_z^2 & \omega_y \omega_z - \dot{\omega}_x \\ \omega_x \omega_z - \dot{\omega}_y & \omega_y \omega_z + > \dot{\omega}_x & -\omega_x^2 - \omega_y^2 \end{vmatrix} \vec{c} $$

The linked post in question also has this useful diagram:

This paper here has a much better summary of the maths used for estimating Center of Gravity using accelerometers: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4208239/

This has the exact equation seen in the quoted section listed as equation (14), with the following line on how to efficiently solve it:

Equation (14) can be solved using a QR-Decomposition based Weighted Recursive Least Squares (QR-D based WRLS) with Forgetting Factor (FF) and covariance matrix resetting threshold (TH).

It then follows through this process in quite some detail, so I suggest reading through their summary.

Answer 2

So I've accepted that I need to correct my accelerometer readings for the accelerometer offset from the center of mass.

The premise is wrong. There is no need to correct accelerometer readings.

• Rockets rotate slowly, if at all. After takeoff, it takes several minutes to pitch over from vertical to horizontal and the acceleration that results from this slow spin is completely negligible compared to aerodynamic and other random forces flight control has to deal with.

• During an engine firing in space, spacecraft generally maintain a fixed attitude (or constant spin with thrust along the spin axis) because everything else is a giant waste of propellant. The burn is timed either with a stopwatch, with a fuel meter, or with an integrating accelerometer. Since the vehicle is in a fixed attitude with zero rotation, accelerometer readings are always correct.

• While there is a need to align the thrust vector with the center of gravity, this is generally done with rate gyros in closed loop control. The engine gimbals are continuously adjusted to keep the vehicle in a fixed attitude, as measured by the gyros. The controller must take into consideration changing mass and center of gravity, but a rough estimate or even a fixed schedule is good enough. For authoritative information about the Saturn V, see How does rocket guidance deal with changing mass?

• While acceleration data is neccessary for inertial navigation systems, there is no need to do any corrections. In fact, such corrections would be harmful, because acceleration data should be sampled as close to the gyro as possible, with both sensors mounted on a rigid platform that is decoupled from the vehicle's vibrations and bending. If one wants to determine the exact velocity vector of a spin-stabilized spacecraft for trajectory purposes, it is sufficient to average the velocity output of the INS over a full revolution while no thrusters are firing.

The answer to the title question "How to correct accelerometer readings?" is: Accelerometers are not even used for primary flight control, only gyros are, and where they are used there is no need for correction, and where there is a need for correction the solution is to maintain a fixed or slowly changing attitude, using gyros and closed loop control.