# Extended Kalman Filter Robotics   Math

## Review of Kalman filter

Distribution of Gausion is not Gaussian, it becomes non-linear

Extented Kalman filter uses a linear approximation of h(x) Here we use first order taylor expansion to

Given a function f(x), a taylor series expansion could be expressed:

$$f(x) \approx \frac{\partial{f(\mu)} }{\partial{x}}(x - \mu)$$

## Design Kalman Filter for 1D tracking problem

We need to define two linear functions: 1. state transition function 2. measurement function

### State transition function

$$x’ = F * x + noise$$

where,

$$F = \begin{pmatrix} 1 & \Delta{t} 0 & 1 \end{pmatrix}$$

$$x = \begin{pmatrix} p \ v\end{pmatrix}$$

postion $p$ is linear motion model, calculation is:

$$p’ = p + v * \Delta{t}$$

Thus We can express it in a matrix form: