Monday, September 18, 2017

Inertial Measurement Units (IMUs)- Some key issues - II

As I indicated in the previous post, the ability to make a "good" IMU is challenged by two basic issues

1) The ability to machine precise parts - such as perfect spheres.
2) The ability to correctly model the behavior of gravity along the IMU's trajectory

The absence of perfect machining creates avenues to add error to the measurement of gravity. This affects the precision of the IMU.

The inability to properly model the local behavior of gravity leads to misinformed notions of accuracy.

There can be a "sweet spot" where acceptable levels of imprecision and inaccuracy coexist in harmony. Under such circumstances, it may be possible to make an IMU that is "good enough" for a particular role. Typically short range ballistic missiles can get away with having "crappier" IMUs simply because they aren't going very far or very high or very fast. However as you get up in speed and altitude, IMUs become quite critical to success.

I suspect the North Koreans are in such a "sweet spot", but I fear they will not be able to stay there very long as their ambitions grow with each passing day.

Here are some ways in which to manage the error in the measurements

1) Comparing measurements on two or more IMUs - If we mount two IMUs on our rocket then we can examine how they differ in the estimates of the height they report. If one IMU is much more sensitive than the other (i.e. able to see differences in height of centimeters as opposed to meters), we could see if it reports a change of 100 units when the coarse IMU reports a chance of 1 m. This kind of thing is pretty common in other measurements. In the published literature you hear words like "Allan Variance" [see this link for more], this refers to way of comparing the performance of two sensors and getting some meaningful measurements of the ARW and drift. In practice, placing multiple IMUs (especially ones with combinations of fine and coarse measurements) on operational platforms is a major manufacturing burden.

2) Error modeling - Once measured there are ways in which one can model the ARW and drift errors in our IMU. Error analysis tools have evolved significantly over the decades. Some really amazing stuff is now available. Most of the methods some variation of "quaternion based filtering". "Quaternions" are a very mathematically compact way of representing the information typically obtained from an IMU. "Filtering" because you are removing noise from the IMU data. Here the community of signal analysts broadly splits into two groups - the DSP guys (who use deep understanding based ideas like Kalman filtering) and the Deep Learning guys (who use techniques like neural networks). It is not clear if either approach gives a clear advantage in terms of accuracy however both approaches require concurrent development of embedded computation systems. That adds large overheads to the manufacturing burden associated with IMUs. You are basically adding a dedicated fab line, firmware development and software validation & testing to the program cost here.

3) RF ranging and other external referencing -You can always use a simple RF signal to correct the accumulation of errors in the IMU. However for extremely long range trajectories, the RF signals run out of line of sight with your rocket. So you have to either use a satellite or do something quite complicated to "get your bearings". If you decide to use a satellite RF beacon, you need to build a really good way of keeping that satellite in a particular spot in space otherwise you can't range off it in any error free way. That part can get really entertaining given all the weird drag effects you have in earth orbit and those gravitational effects I alluded to earlier. Also you are now adding the cost of a satellite beacon program to the cost of your rocket guidance program. This quickly devolves in to a number of chicken-and-egg questions. A highly unpleasant situation but sometimes a way can be found. One of the my favorite ideas in this context is Stellar Navigation. A combined "Astro Inertial Navigation" system was used on the SR-71 and a stellar alignment system was used on Gravity Probe B. These are relatively simple to implement and very robust. The unfortunate side effect of external referencing is that it can be interfered with and that makes its less suitable for nuclear deterrence missions.

In my next post I will discuss some novel gravity sensing systems that are finding application in commercial IMUs and how things might play out for them in the future.

(cont'd in next post).


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