Where the velocity the object time and modeling error term

Section 1 provides a short introduction and motivation for the Kalman filter. In Section 2 you will use the concepts seen during the online videos and live sessions to develop the equations of one Kalman filter. The deliverables (including the implementation of the filter using Matlab) are finally presented in Section 3.

1 Introduction

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axis is discretised uniformly, with an arbitrary stepsize of 1 time unit. The tracking problem then starts at t = 1.

This project contains 12 questions, with a total of 36 marks. This project counts for 30% of the final mark.

2 Theory: Near constant velocity (NCV) model (1D)

where v(t)is the velocity of the object at time (t) and n(t) is a modeling error term, assumed to be zero-mean Gaussian with variance σ2 z. Note that here we assume ∆t= .01 (we receive a measurement every 10ms) and that n(t) is assumed to be independent of the other terms in Eq. (1).

Question 1: Under these hypotheses, derive the expression of the probability density function f(z(t)|z(t−1), v(t−1), σ2 when conditioning on z(t−1)and v(t−1), these variables are assumed to be known. (2 Marks) z) and explain your reasoning. What are its mean and variance? Note that

p(t)= Qp(t−1)+ n(t), (3)

where p(t)= [z(t), v(t)]Tand n(t)= [n(t) z, n(t) v]T are two 2 × 1 vectors and Q is a 2 × 2 matrix.

In practice however, R can be user-defined. For instance, one could choose

In the remainder of this project we will use Eq. (4), unless stated otherwise.

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where n(t)denotes the observation noise at time t.

vector p (position and velocity here). First, let us introduce some variables

• mt|t−1: Expected value of p at time t, before observing y(t)(only y(t−1)= [y(1), . . . , y(t−1)] have been observed).

• mt|t−1 and Σt|t−1 are respectively the mean and covariance matrix of f(p(t)|z(t−1)).

• mt|t and Σt|t are respectively the mean and covariance matrix associated with f(p(t)|z(t)).

First we will predict the value of p(1)(before observing z(1)) using the result of Question 3.

Question 6: using Eq. (3), verify that f(p(1)) is the density of a bivariate Gaussian distri-

Question 7: If we assume the f(p(t−1)|y(t−1)) is a Gaussian probability density function and using Eq. (3), verify that f(p(t)|y(t−1)) is the density of a bivariate Gaussian distribution and compute its mean and covariance matrix. Detail how you obtain the final result. (2 Marks)

So far, we have shown how to perform the prediction steps if f(p(0)) is a Gaussian distri-

Question 10: By identifying the terms that are linear and quadratic with respect to p(t)in

the exponential factor of f(p(t)|y(t)), show that the mean and covariance matrix of f(p(t)|y(t)) are given by

and I2 the 2 × 2 identity matrix. (6 Marks)

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Algorithm 1

At the end of the project, you are asked to submit a single pdf file

• B39AX Project202122 XXXX.pdf

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are. The file ”data1.mat” contains a sequence of T = 104measurements and time instants (see template). The file ”data2.mat” has the same structure but some of the measurements have been corrupted (to mimic missing data). Before processing the data using the Kalman filter that you will implement, add the last two digits of your HWU ID to the vectors y0 and y to generate new ground truth and noisy data (see lines 14-17 in the *.m file). Your code should include a step that controls if the data is corrupted (NaN) and adapt the filtering step accordingly. When including your code in the report, make sure that it can be run to display the four figures requested below.

Guidelines for figures: Always include a title and xy labels (including units if available). If you plot more that two curves or sets of points, add a clear legend.

Question 12: If you observe a difference between the estimated variances for data1 and data2, explain why it happens. (2 marks)

USA: Cambridge University Press, 2013.