On the bias and flicker noise of inertial sensors

What is bias anyway? Commonsense says that it is the sensor output when the input is 0. Therefore, if you align your gyroscope to the East and read 2.5434V at the output, you may think that you can call this value as your gyroscope bias. But, will this value be your real sensor bias? Is the commonsense right?

Let us see what we have in our hands. The output of every inertial sensor consists of at least 4 components:

  1. Real acceleration/rotation rate: we can be sure that these are 0 during the bias tests by properly aligning our sensors.
  2. Additive white noise: we can be sure that the effect of this is 0 by computing sufficiently long averages rather than measuring a single sample.
  3. Bias: this is what we try to estimate
  4. Flicker noise: this is the noise that puts a lower limit to the Allan variance.

Every time you measure the sensor output, you must always assume your readout is at least the sum of bias and flicker noise. This brings us another important question. What is a flicker noise? Can we get rid of it by again computing long averages of the sensor outputs?

There are plenty of papers out there which investigates the physics behind the flicker noise in the semiconductors.  It could be quite difficult to understand them thoroughly. However, as a navigation engineer you are at least supposed to know following basic facts about the flicker noise even if you do not understand it:

  1. The power of the difference between 2 samples of the flicker noise (i.e. y(T)-y(0)) increases with time proportional to “2h.ln(T)” where h is the flicker noise value.
  2. The power of the average of this difference also increases with time proportional to “h.ln(T)”. In other words:

This 2nd property is extremely important for us to understand the concept of bias in low cost inertial sensors. There are significant number of people out there who are not even aware of this simple property, yet do not hesitate writing papers (and even books) on INS. Naturally, these texts written by such idiots contain lots of statements which directly contradict to above properties of the flicker noise.

So, what should these properties mean to you? Let me list a couple of important results so that you do not repeat the same mistakes in your future papers as those fools did.

  1. Property 1 dictates that the flicker noise itself is divergent. This means that the value of the flicker noise can theoretically be anything between -infinity to +infinity.
  2. Property 2 dictates that the average of the flicker noise is also divergent. This means that you cannot eliminate the flicker noise by taking the average of sensor outputs regardless of the averaging duration.
  3. (But, we have to define a bias value. We cannot throw away an inertial sensor just because we cannot come up with a theoretically consistent bias estimate.We are engineers, not physicists.) Let us suppose that we define the sensor bias as the average of the sensor outputs in T seconds. Property 2 also dictates that this bias value changes more as you increase T. In other words, if you plot time vs average value (your bias estimate), you will see that your bias estimate changes more with time (and eventually diverge) instead of converging to a fixed point.

You must always remember this 3rd result. It simply says that it is impossible to define a constant bias value for an inertial sensor regardless of the averaging duration. In other words, the longer the averaging duration is, the more diverse bias estimate you will observe. You should not be tricked by the fact that the Allan variance of flicker noise is constant regardless of the cluster time. Although the Allan variance is constant, the power of the flicker noise average grows with the averaging duration. This is in fact quite an interesting result. As an example, suppose you compute 2 different bias values by taking the average of sensor outputs for 5 minutes and 1 hour respectively. The variation (the power of change) of 1-hour bias estimate will be bigger than the 5-minutes estimate. However, this does not mean that one of these bias estimates is better than the other. They are just 2 different values which are equally valid (or equally wrong). This is because there is no constant value that you may call as bias. You cannot differentiate flicker noise from bias. You must assume that the bias itself evolves in time in an unpredictable manner.

As a result, there is nothing called perfect bias calibration. You cannot claim that the bias calibration value that you compute now can be assumed valid some later time. The best bias estimate is the one that is computed most recently, not the one that is computed with longer averaging durations. As a matter of fact, you should consider bias as the current state of the flicker noise. As the flicker noise of low cost MEMS units are very dominant, representing the bias as the state of the flicker noise is a better mathematical model than random constants.

The next time you see an “INS expert” claiming that he/she obtain better bias estimates by repeating the calibration tests with longer times (or with more position), you can now easily conclude that he/she is another self deceptive ignorant that you should stay away.

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