In this video, we're going to have a look at a like measurement model.
Now, this is going to be slightly different to the linear model, such as the GPS model that we looked
at inside the linear coming filter, because this lighter model is going to be a nonlinear model.
So to implement this model, we're going to have to use a non-linear filter, such as the extent and
filter.
So to recap what we did for the linear, come and fill it up, we had a GPS measurement model.
So basically the model produced a measurement of the position of the vehicle.
So it produced IPEX in a P position.
So for the state vector that described the vehicle in the linear frame, we had the position X, Y and
then we had the velocity in the X and Y.
The measurement vector basically gave us a measurement of the position of the vehicle in the X and Y,
and it was a linear model such as it could be written in this form here.
So expanding out the measurement vector, the position, the X and Y was just basically a modification
of this election matrix C, which selects out the position X and the position in the Y from the state
vector, and it also had additive noise added onto it.
So this was for japes like SANZAR directly measuring the position of the vehicle, which just directly
pulled out the position from the state vector.
So now let's consider the non-linear 2D range and bearing measurement model.
So this is going to be a type of measurement that gets returned by a law to model.
So let's consider we have a vehicle here.
So it's located at position X and Y.
It has a heading of PSI and also has a velocity V in the direction of the heading.
But this is not going to be important for this measurement model.
So the measurements that it produces, it produces a range measurement.
So the range between the vehicle location and the landmark location.
So we can say that we have a landmark here.
This is going to be the position of the landmark and the X and Y for Landmark I.
So we're going to get a range I, which is going to be the range from the vehicle to Landmark.
We're also going to get a relative bearing.
So this is the bearing between the heading of the vehicle and the landmark direction.
So this is going to be denoted as our Theta I.
So the state picked up for this non-linear model is going to be, again, a velocity heading of the
vehicle and a position in the X and Y, the measurement vector.
So these are the measurements we're going to get from the Leida model are going to be the Range I and
Theta I.
This is going to be the range and relative bearing to the landmark inside the visible range.
So we're going to get Modibo measurement vectors for each of the landmarks that the LIDAR model can
see.
And the measurement model is going to be represented as this nonlinear model here.
So we're going to have the nonlinear measurement Model H, which is going to be a function of the current
state, the current noise, and it's going to produce the current measurement vector.
Now, this measurement model can be risen like this.
So basically the first component is going to the range.
So we can say here we get the difference between the landmark location and the vehicle location and
the X squared, plus the landmark location and position of the vehicle in the Y squared.
And then we get the square root of this.
So this is just Pythagoras Theorem.
It's just going to get the distance between this point and this point, the difference between the vehicle
location and the landmark location for the landmark.
Down here, we're going to get the relative bearings.
This is basically going to take the angle between this point and the landmark location.
This is going to give us this tunnel angle here.
So this is the Eytan component down here.
So that's going to give us the angle between the X axis and the landmark.
However, we want the relative bearings.
So as the heading of the vehicle changes, we also want to take into account how that changes its relative
bearing.
So if we have this tunnel angle here to the landmark and we subtract from this angle our heading, we're
going to get left with this relative bearing.
So this is going to be input into the system.
So this is going to be our nonlinear measurement model for our little sensor.
So the measurement vector is going to be a normal function of the state vector and we're just going
to have some additive noise.
So this is a lot of like sensor.
So it measures the relative position in polar coordinates.
And this is going to be important because using this model here, we get information about the position
of the vehicle, but we also get information about the heading of the vehicle as well, because the
heading becomes part of the measurement model.
So this is going to be a more powerful way of estimating what the state of the vehicle is as compared
to the GPS sensor, the GPS sensor.
And he gives us information about the position of the vehicle while this sensor gives us information
about the position of the vehicle.
But it also gives us information about the heading of the vehicle.
So we can see later on that this has the potential to dramatically increase the precision of the vehicle
state estimates.