This is an old revision of the document!
To understand state space models, and in particular, the Kalman filter.
For this lab, you will implement the Kalman filter algorithm on two datasets. Your notebook should produce two visualizations of the observations, the true state, the estimated state, and your estimate of the variance of your state estimates.
For the first dataset, you will implement a simple random accelerations model and use it to track a simple target. Your final visualization should look something like the following:
Your notebook will be graded on the following elements:
For the first dataset, you will perform simple target tracking using the random accelerations model. You should use the equations given in Lecture 17. The data can be found here:
There are two arrays in this .mat file: data
contains the actual (noisy) observations at each timestep, while true_data
contains the true x,y coordinates at each timestep. Note that your kalman filter should only use the noisy observations; the true x,y coordinates are only given for visualization purposes.
To visualize the uncertainty at each timestep, the code for plotting the 95% confidence interval of a Gaussian, from lab 6, can be used.
The model is given by
# our dynamics are described by random accelerations A = np.asarray([ [ 1, 0, 1, 0, 0.5, 0 ], [ 0, 1, 0, 1, 0, 0.5 ], [ 0, 0, 1, 0, 1, 0 ], [ 0, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ] ]) # our observations are only the position components C = np.asarray([ [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]]) # our dynamics noise tries to force random accelerations to account # for most of the dynamics uncertainty Q = 1e-2 * np.eye( 6 ) Q[4,4] = 0.5 # variance of accelerations is higher Q[5,5] = 0.5 # our observation noise R = 20 * np.eye( 2 ) # initial state mu_t = np.zeros(( 6, 1 )) sigma_t = np.eye( 6 )
To avoid numerical instabilities in the calculation of your covariance matrices, you may want to use np.linalg.pinv
.