ukf2d_written_solutions.tex

\documentclass{article}
\usepackage{fullpage}
\usepackage{amsmath} % for math stuff
\usepackage{graphicx} % for inserting graphics
\usepackage{xcolor} % for text coloring

\newcommand{\todo}[1]{\textcolor{red}{\textbf{#1}}}

\title{UKF with 2D State Vector}
\author{}
\date{\today}

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\begin{document}
\maketitle

\section{Design}

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\subsection{State Vector}

\begin{equation}
  \mathbf{x}_t=\begin{bmatrix}
  z \\
  \dot z
  \end{bmatrix}
\end{equation}

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\subsection{Prediction Step}

\subsubsection{Control Input}

\begin{equation}
  \mathbf{u}_t = \begin{bmatrix}
  \ddot z
  \end{bmatrix}
\end{equation}

\subsubsection{State Transition Function}

\begin{equation}
  g(\mathbf{x}_{t-\Delta t}, \mathbf{u}_t, \Delta t) = \begin{bmatrix}
  \todo{?} & \todo{?} \\
  \todo{?} & \todo{?}
\end{bmatrix}\mathbf{x}_{t-\Delta t} + \begin{bmatrix}
  \todo{?} \\
  \todo{?}
  \end{bmatrix}\mathbf{u}_t
\end{equation}

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\subsection{Measurement Update Step}

\subsubsection{Measurement Vector}

\begin{equation}
  \mathbf{z}_t = \begin{bmatrix}
  r
  \end{bmatrix}
\end{equation}

\subsubsection{Measurement Function}

\begin{equation}
  h(\mathbf{\bar x}_t) = \begin{bmatrix}
  \todo{?} & \todo{?}
  \end{bmatrix}\mathbf{\bar x}_t
\end{equation}

\subsubsection{Measurement Covariance Matrix}

Populate the measurement covariance matrix $\mathbf{R}_t$ with a reasonable value for the variance $\sigma^2_r$ of the range reading $r$.

\begin{equation}
  \mathbf{R}_t = \begin{bmatrix}
  \todo{?}
  \end{bmatrix}
\end{equation}

\noindent \todo{Your histogram image here}

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\section{Implementation}

\subsection{Initializing $\dot z$}
\todo{Your answer here}

\subsection{Consecutive Predictions vs. Updates}
\todo{Your answer here}

\subsection{Comparison of UKF to EMA Filter}
\todo{Your answer and Height Readings chart image here}


\end{document}