Estimating Scalar Value from Multiple Observations

Question

I'm working on a project in which I get data from ~15 same sensors located differently around the object and I want to estimate the state of the the object I'm intersted by using multiple observations. Therefore I want to use Kalman filter for estimation. But I'm not very clear at this point because i saw scalar value estimation from scalar observation (like in the temperature example) or vector estimation from observation vector (like 2D or 3D position estimation) but i never saw scalar estimation from a observation vector? Books and lecture notes give always these two examples. Is there any example/book/article that may help to me? Thanks

Answer 1

In your case you need to interpret the observations in another way.

Let's say you need to estimate a scalar position of an object in 1-D space (the X coordinate). For this you have 15 sensors (GPS, odometry or something else). Each sensor gives you a measurement of X with some uncertainty value (variance from the datasheet).

To keep it simple let's your state vector incorporates only the X position. Your measurement consists of 15 values each of them can be seen as a separate measurement. So your measurement model would incorporate only one value (with variance), but you need to go through all the 15 observations.

Your Kalman Filter process would look like this:

init()
update(sensor1)
update(sensor2)
...
update(sensor15)
predict()
update(sensor1)
update(sensor2)
...
update(sensor15)
predict()
...

This approach is very flexible:

• your model does not depend on the number of sensors. if at some point in time you have only 12 measurements, you just perform update 12 times.
• your matrices are really small (a scalar is a 1x1 matrix as well)
• the sequence of the updates does not matter, the fusion result will be the same for updates 1 to 15 or 15 to 1 or what ever.

Have a look at this post How to use a Kalman filter with two irregular observations of the same variable?

It shows a very similar problem with 2 measurements in a 2-D space.