Kalman Filter Prediction Implementation

I am trying to implement a Kalman filter in order to localize a robot. I am confused with the prediction step (excluding process noise) x = Fx + u

If x is a state estimation vector: [xLocation, xVelocity] and F is the state transition matrix [[1 1],[0 1]], then the new xLocation would be equal to xLocation + xVelocity + the corresponding component of the motion vector u.

Why is the equation not x = x + u? Shouldn't the predicted location of the robot be the location + motion of the robot?

Maybe there is some confusion with respect to what the matrices actually represent.

The "control vector", u, might be the acceleration externally applied to the system.

In this case, I would expect the equations to look like this:

x_location = x_location + x_velocity

x_velocity = x_velocity + u_velocity

These two equations assume that the update is applied every 1 second (otherwise some "delta time" factors would need to be applied and included the transition matrix and the control vector).

For the situation mentioned above, the matrices and vectors are:

• The state vector (column vector with 2 entries):

  x_location

  x_velocity

---

• The transition matrix (2 x 2 matrix):

  1 1

  0 1

---

• The control vector (column vector with 2 entries):

  0

  u_velocity

---

This link contains nice explanations and visualizations for the Kalman Filter.