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

    1.5 Simulation (cont'd...)

    Measurement model

    The measurement vector of a particle is denoted by $Y_{k}$ which is related to actual state vector $X_{k}$. The measurement model of a moving particle in vector form at step $k$ is \begin{equation} Y_{k}=H X_{k}+v_{k}, \end{equation} where, $H_{k}$ is the measurement matrix and $v_{k}$ is the measurement noise, which is white Gaussian. Where, process noise vector and measurement noise are independent and uncorrelated to each other to each other. We can measure position, velocity and acceleration of a particle then we use $H_{k}$ is a $3\times 3$ identity matrix. If we measure only position and velocity then we use $H_{k}$ is a $2\times 3$ matrix, which is a sub-matrix of a $3\times 3$ identity matrix. If we are measuring position and velocity then the measurement model of a moving particle in matrix form is \begin{eqnarray} \begin{bmatrix} y_{1_{k}} \\ y_{2_{k}} \end{bmatrix} =\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_{k} \\ \dot{x}_{k} \\ \ddot{x}_{k} \end{bmatrix} +\begin{bmatrix} v_{1_{k}} \\ v_{2_{k}} \end{bmatrix}, \end{eqnarray} where, $y_{1_{k}}$ measuring the position and $y_{2_{k}}$ measuring the velocity of a particle. The measurement noise covariance matrix $R_{k}=E[v_{k}v_{k}^{T}]$. Note that measurement vector $Y_{k}\in \mathbb{R}^{m}$, measurement matrix $H_{k} \in \mathbb{R}^{m\times n}$, measurement noise $v_{k} \in \mathbb{R}^{m}$ and measurement noise covariance $R_{k} \in \mathbb{R}^{m\times m}$.

    Simulation results

    We should show the computer simulation result for tracking the target at different value of standard deviation $\sigma$, where the target acceleration is known. We are measuring both position and velocity of the target for every 10 millisecond intervals i.e. T=0.01 seconds. We assume that measurement errors have a normal distribution with zero mean, specified covariance and also independent from sample to sample. The standard error of position was taken to be 100 m, and that of velocity measurements was 4 m/s. The measurements were fed to a Kalman filter, which then estimated the target motion from the simulated measurements. The filter state was initialised to the correct value $X_0=[0 \; 0\;0]^T$ and the initial predicted covariance matrix was set to be $P_{0|0}=diag[10 \;\;1 \;\; 0.1]$. The error of position, velocity and acceleration have been plotted in figure 3.

    Truth position, velocity and acceleration
    Figure 3. Error of position velocity and acceleration estimation $\sigma=0.003$

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