# Multiple cheap IMUs vs one expensive IMUs

In my project, there is a possibility of either buying one expensive IMU or 5 cheaper IMUs. They have to be cheaper due to budget limitations, and the idea is to use them in order to be able to do "voting" - i.e. vote out outlier measurements of one IMU if 4 other IMUs show a different reading - and also for redundancy. Question: does anybody know if 5 cheaper IMUs (on the order to 300 dollars each) can create a signal at least as smooth, via sensor fusion, as one expensive IMU (on the order of 1500 dollars)?

• what accuracy do you require? – geometrikal Oct 29 '15 at 10:16

That really will depend on the variance (standard deviation) of the measurements you get from the five vs the one.

Suppose your expensive IMU gives a measurement distributed as: $$x_\mbox{expensive} \sim N\left(x_\mbox{truth}, \sigma^2_\mbox{expensive}\right)$$ and suppose your cheaper ones give measurements distributed as: $$x_\mbox{cheap}^i \sim N\left(x_\mbox{truth}, \sigma^2_\mbox{cheap}\right)$$ for $i=1\ldots 5$ and that you form the average of these cheaper measurements to get the "real" measurement: $$\hat{x} = \frac{1}{5} \sum_{i=1}^5 x_\mbox{cheap}^i$$ then $\hat{x}$ is distributed as: $$\hat{x} \sim N\left(x_\mbox{truth}, \frac{\sigma^2_\mbox{cheap}}{5}\right)$$ So if $$\frac{\sigma^2_\mbox{cheap}}{5} \le \sigma^2_\mbox{expensive}$$ you should be OK with the cheaper sensors.

Of course, there may be other problems with your cheaper sensors: they may be biassed (so that the mean of their distributions will all be different from $x_\mbox{truth}$ ).

And I'm not quite sure what you mean by "voting"? Do you vote off the one furthest from the mean of the 5? What if the mean changes for the remaining four? I'll do a bit more digging to see if there's a viable algorithm there, but the direct mean of all five might be the easiest.

• By voting I mean if one of the IMUs fails or temporarily gives measurements very different from the other four, then a "vote" will be taken by the main computer to decide if the other IMU is to be trusted (based on statistics/probability of it obtaining such values). Then the other IMU will be neglected and only four measurements used, which gives robustness to the setup. I'm hoping to use sensor fusion, which I hope will give a better result than the $1/5$ that we obtain with just an average. – space_voyager Oct 29 '15 at 10:11
• @space_voyager : Thanks for the info! In order for sensor fusion to do any better, you'll need to have some sort of a model as to how the $x_\mbox{truth}$ values change. Then you can set up a Kalman filter or some other estimator, and get a better variance. Do you have such a model, or will a simple 1D (2D? 3D?) model do? – Peter K. Oct 29 '15 at 10:37
• Yes, I have such a model. – space_voyager Oct 29 '15 at 10:42
• @space_voyager : Can you share it? Is it amenable to application of the standard KF equations? – Peter K. Oct 29 '15 at 11:44
• I cannot share it unfortunately because I don't have the model yet... but I know it exists and, yes, it look like it'll be amenable to application of standard KF equations. I'd like to ask what may be a difficult question. I've had so far good exposure to statistics, math and optimal control, but close to nothing on estimation. I'll have a course next semester on it, but I need to apply this to my project before the course finishes. Could you suggest a solid reference that is readable over 2-3 days that will give me a solid foundation with which I'll be able to understand papers? – space_voyager Oct 29 '15 at 12:49

If you are going for the 5 IMU option, you should definitely use Kalman filtering with them. It is extremely easy to implement with as many IMUs you like. If you do not understand the whole concept yet, I suggest watching the following playlist. Special Topics - The Kalman Filter

Very down-to-earth lectures and if you write everything down from his whiteboard, you can just "C&P" the mathematical formulas to your code and fine-tune from there.

• Welcome to DSP.SE. In the future, please try not to be offensive or rude in your answers. – jojek Jun 5 '16 at 14:54