Monday, September 18, 2017

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

The proliferation of Inertial Measurement Units (IMUs) has rightly caused people to become concerned about the likelihood of their misuse by rogue states. There are however physical constraints that limit certain kinds of misuse. I discuss some of the key limitations below.  A good reference to have handy for this is "Inventing Accuracy". If you have problems following what I am saying, please reply to this post in the comments below and I will get back to you asap.

For the purposes of this discussion, let us consider a simplified IMU which consists of a gyroscope and a gradiometer. The gyroscope ensures that the gradiometer is aligned with vertical direction. In our simple model, the gyroscope is a mechanical device- a spinning wheel (the kind you might find in an undergrad physics lab) and the gradiometer is a simple spring which is compressed/stretched by a test mass attached to it. Also let us assume that our IMU is non-ideal in predictable ways and that our IMU is attached to a rocket that behaves in a totally predictable way (these are both over simplifications that do not hold IRL).

In the ideal case, our gryoscope is spun up to a certain angular velocity about its vertical axis and since the entire assembly sits on a gimbal mount, it holds the spring and test weight of the gradiometer perfectly vertical.  The test mass experiences a gravitational field that pulls it downwards and this causes the spring in the gradiometer to extend. If we apply an acceleration to the IMU (as we might if we were to light the rocket engine under it), we see the extension change as the added acceleration also pulls on the test mass.

In the ideal world, our IMU works perfectly, as the rocket engine lights up we see added acceleration add to gravity and the extension increases. As the rocket rises into space, the acceleration due to gravity reduces. A computer attached to the IMU records the change in the extension with time and when the change in extension reaches a particular amount, the computer attributes this change to the rocket reaching a particular height above ground and shuts off the rocket engine. Everyone is happy.

That's not the way it works IRL.

Firstly our gyro experiences friction on its bearings. This leads to a torque that changes its angular momentum. The decline in angular momentum presents in two ways - firstly as a set of random angular deceleration events that cause the angle of the gyro to rattle around (this is called Angular Random Walk or ARW) and secondly as a slow reduction in its angular velocity that causes the angle of the gryo to shift in one direction (this is called "drift"). As the gradiometer is attached to gyro, shifts in the gyro angles propagate to the measurements of acceleration. The exact model of propagation is quite nontrivial but in this way the gradiometer picks up an ARW and Drift of its own.
Errors in the gradiometer reading (i.e. extension) translate into errors in the estimation of the height of the rocket above ground. A large error could significantly alter the trajectory of the rocket.

A mechanical gyro and gradiometer may sound very low tech, but they are based on technologies that are over a hundred years old. They are extremely reliable. If you can machine perfect spheres (turns out that is a lot harder than one might think it is) you can make very high precision and high "accuracy" IMUs. I use "accuracy" in quotes because it turns out that it is quite difficult to define the term in this peculiar context.

As we go up and out from earth, we experience gravitational contributions from poorly characterized terrestrial (such as the non spherical nature of earth) and extraterrestrial sources (the moon, nearby asteroids, tidal effects etc...). These effects make it hard to claim deep knowledge of the gravitational acceleration at various altitudes. This makes it difficult to define "accuracy" in the context of a gradiometer.

(cont'd in next post).


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