We have already covered that no sensor is perfect and that they can return different types of errors.
Now, sensors themselves can also return erroneous data or faulty data.
And this can be for a number of reasons.
It could be a mechanical issues like the sensor could have come loose or could be a hardware issue.
The sensor was broken or it could just be the characteristics of the sensor.
It could be such like in GPS.
If we lose satellite signals, the GPS data is not going to be reliable, is no longer going to give
us a valid position measurements.
So there's many different reasons why the sensor itself might give us faulty data.
So sensors and measurements are not perfect.
We already assume that the measurements that they provide are corrupted with noise.
They can also be cases when the measurements are completely erroneous and outside the noise profile
due to hardware or sensing faults.
As we've discovered, fusing wrong information into the common filter can cause a common field to diverge
and degrade the performance.
And a single wrong measurement could be enough to make the filter unusable for the rest of the time
it's running.
So this is very important that we don't use any wrong information into the common filter.
We want to stop the faulty measurement from being used in the first place.
And this is pretty much the only way to deal with this type of faulty data.
And this process is commonly called Foch detection and isolation.
So we wanted to detect when a sensor is faulty, that the data is faulty and we want to isolate it from
the rest of the system before it allows this incorrect measurement to corrupt the system.
So there are a number of different typical sets of thoughts that you see inside the sensor, the first
one we're going to look at is when we get erroneous spikes.
So pretty much as time goes on, we might get these large jumps in data.
So you get these spikes every now and again and they're totally out of line with what the sensors should
be reading.
So you can sort of clearly identify these are erroneous measurements, another type of sense of what
is called a lock in place for.
So we might have the ideal sensor producing a measurement like this, but at some time it locks in place
and it produces a constant output afterwards, even though the measurement should be producing this
kind of line up here.
So this could be so this is more typically of hardware fault.
So if something breaks inside the system, say a linkage breaks to something or something comes loose,
the sensor is going to get disconnected from what it's sensing.
It can stay locked in place.
Another typical sense of what is called a hard over.
So basically we have initially the sense is following the truth quite well until we get to a certain
point and then the center basically becomes saturated.
So it goes all the way up to the max or it could go all the way down to the minimum.
And again, this is typically usually another how if something might have come loose, causing the sense
I just to diverge.
So in a similar point of view, we might have a dead center.
So this is basically as a sensor is running something like in a loose wire or something, something
just disconnects and then we just get a zero output for the rest of the run.
And then lastly, another typically fault that you get when a sensor disconnects is called a floating
error.
So the sensor is tracking well, and normally up until this point and off this point, suddenly the
output becomes random and basically starts to float around.
So something could have come loose or something could be flapping in the wind.
It could be a loose connection on the wire and it's just jumping around all over the place.
So these most of these errors here can be caused by hardware issues or mechanical issues, or it could
be loose connections or something like that.
So with the spike faults, what we want to do is make sure we don't fuse these erroneous spikes inside
it for the rest of these forces.
Basically, once we've detected that the fault has occurred, we can't trust the sensor anymore.
So once that happens, we can't use any of the measurements from the sensor after the time that the
fault has occurred.
So we want to isolate it and or for the rest of the time, whereas with the spikes, we can probably
just ignore that single measurement or number of measurements that are no longer consistent with what
we expected the sensor to be outputting.
The main way to detect Sensa faults is from the measurements themselves, it is to look at the measurement
innovation from the carbon filter or some statistical probability based on the innovation.
And this is to say if the sensor is performing as specified or as expected.
If the sensor is detected as 40 or the current measurement is bad, then we want to ignore the data
rather than just fusing it into the common filter and corrupt the estimates.
So the main way to do this is called innovation checking so we know the probabilistic properties that
the innovation should follow.
And we've covered this in common filter.
We want the innovation should be zero mean, so we want to drive the innovation desire to drive the
estimated errett is are.
We also know the innovation covariance and this is estimated in the first part of the coming forward
update step.
So this is The Matrix.
So the innovation here should be a random variable from a Gaussian distribution with zero, a mean and
covariance s.
So using this probabilistic model of the innovation, we can come up with some innovation checking criteria.
We can use what we know about probability to test new measurements before we use them into the filter.
So if we have a look at the one, the innovation, normal distribution.
So basically we're looking at this normal distribution here of main zero covariance and this is the
innovation here.
So we know that the innovation should have zero mean and it has a Gaussian or normal curve that looks
like this with covariance s and we know from Gaussian or normal probability distribution that anything
within plus or minus one standard deviation from the main encapsulates point six, eight or 68 percent
of probability that the value of the random variable should fall into this range.
And then similarly for two sigma, we know that it should encompass this 95 percent and three sigma
should compas 99 percent.
So if we get an innovation value all the way over here, there's probably less than a one percent probability
that the measurement is valid.
So we can use this criteria to test whether the measurement is valid by looking at the innovation value.
If the innovation value is larger than expected, then we know that is a faulty measurement and this
can be simply tested by a simple comparison operator.
So if the value or size of the innovation is greater than two times the square root of S in this case
here we know that there is less than a five percent chance that the measurement would be valid based
on the two sigma standard deviation.
So basically we can set a threshold and ignore these long values from the innovation.
So basically anything larger in this case than our Three Sigma means is less than a one percent chance
is a valid probability for a valid measurement.
So this works quite nicely in the one dimensional case.
However, what can we do for when we have a multidimensional Gaussians?
So instead of having a single value for the innovation, when we actually have the innovation vector
like we've been dealing with, and to do this, we use something called the normalized innovation squared
or is so for the multidimensional case we can use to normalize innovation squared the normalized innovation
squared, it should follow a unit Gaussian distribution.
We can test for this property using the CHA Square test, also known as the person's CHA squared test.
So now let's let the innovation vector mue be in dimension of innovation vector and let s be the innovation
covariance matrix in normalize innovation squared can be calculated using this equation here.
So basically we calculate the innovation squared and we normalize it by the uncertainty inside the system
to calculate the NSA.
So this this time over here, we can test to see if the innovation belongs to the unit Gaussian distribution
for a specific probability level using the GI Square test.
And this test can be written like so.
So here we have the innovation value here.
There is now for this to be a valid measurement from the ideal distribution it has to meet this criteria
here.
So we have to say the price is less than this value here or this threshold here.
Now, this is the tri squared distribution, so you can see the chai squared Alem here is our degrees
of freedom.
So this is going to be the measurement vector size.
So we have a single measurement is going to be and is going to be equal one if we have like a two dimensional
measurement and is going to be equal to.
And likewise.
And then this value here is the P value.
The pay value here is going to be how certain we want to be of this test, so we normally typically
say I pay value of anything smaller than point zero.
Five is significant.
So what we're saying with this pay value here is that if we do one minus this pay value, in this case
it be 95 percent, we will have a 95 percent confidence that this measurement belongs to this distribution.
If these properties if it's less than this value here, i.e. the measurement is faulty, if the value
is larger than the value of the tri squared distribution.
So the choice squared distribution value can be looked up in tables to find the value of which you want
to set the threshold to, so I listed a few values in this table here.
So this is for the pay level of point zero five and the number of measurement dimensions.
So for a two minute two dimensional measurement vector, so such as if we had the x y position, we
can set a square, a threshold of paying five point nine nine.
This can also be read off over this table in here.
So for a thirty two degree, I'm afraid.
So this blue line here, we can look for a pay level or point zero five, which is all the way down
towards here.
So if the Naess is larger than this value here, then we know that with this historical significance
that the measurement is out of range.
The measurement or innovativeness does not belong to the distribution as expected.
And in this case, we should ignore the measurement.