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## 1. Kalman Filter Tutorial (Cont'd...)

### 1.5 Simulation (cont'd...)

#### Model design

In a simple Kalman filter we assume that a particle moving in a straight line, where, we assume a particle position, velocity and acceleration as a state variable. A particle state vector at step $k$ is denoted by $X_{k}$. The process model of a particle in vector form at step $k+1$ is $$X(k+1)=\phi_{k} X_{k}+w_{k},$$ where, $\phi_{k}$ is the state transition matrix, and $w_{k}$is the additive noise component(process noise), which is white Gaussian. We denote the position $x_{k}$, velocity $\dot{x}_{k}$ and acceleration $\ddot{x}_{k}$ at step $k$. The process model of a particle in matrix form at step $k+1$ is \begin{eqnarray} \begin{bmatrix} x(k+1) \\ \dot{x}(k+1) \\ \ddot{x}(k+1) \end{bmatrix} = \begin{bmatrix} 1 & \tau & \tau^2/2 \\ 0 & 1 & \tau \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x_{k} \\ \dot{x}_{k} \\ \ddot{x}_{k} \end{bmatrix} +\begin{bmatrix} w_{1_{k}} \\ w_{2_{k}} \\ w_{3_{k}} \end{bmatrix}. \end{eqnarray} The process noise covariance matrix $Q_{k}=E[w_{k}w_{k}^{T}]$ is assumed to be known. Note that state vector $X_{k} \in \mathbb{R}^{n}$, state transition matrix $\phi_{k} \in \mathbb{R}^{n\times n}$, process noise $w_{k} \in \mathbb{R}^{n}$ and process noise covariance $Q_{k} \in \mathbb{R}^{n\times n}$. Here $X_{k} \in \mathbb{R}^{n}$ means $X_{k}$ is a real vector which dimension is $n\times 1$. The one-dimensional target motion problem is given below:
$\bullet$ Stationary for 5 seconds from t=0 to t=5 seconds.
$\bullet$ Constant acceleration of 10 $m/s^2$ for 10 seconds from t=5 to t=15 seconds.
$\bullet$ Constant velocity for a further 15 seconds from t=15 to t=30 seconds.
The truth position, velocity and acceleration has been plotted in Figure 2.

Figure 2. Truth position velocity and acceleration