From 39cf255afbce6217a0368cd4c184c095019aa1a0 Mon Sep 17 00:00:00 2001 From: Refik Hadzialic Date: Wed, 12 Sep 2012 17:09:34 +0200 Subject: Appendix correction --- vorlagen/thesis/maindoc.pdf | Bin 8064792 -> 8064835 bytes vorlagen/thesis/src/kapitel_A.tex | 17 +++++++++-------- 2 files changed, 9 insertions(+), 8 deletions(-) diff --git a/vorlagen/thesis/maindoc.pdf b/vorlagen/thesis/maindoc.pdf index 7e0ecee..b638b1a 100644 Binary files a/vorlagen/thesis/maindoc.pdf and b/vorlagen/thesis/maindoc.pdf differ diff --git a/vorlagen/thesis/src/kapitel_A.tex b/vorlagen/thesis/src/kapitel_A.tex index 6ce6cc0..6bc3dc4 100644 --- a/vorlagen/thesis/src/kapitel_A.tex +++ b/vorlagen/thesis/src/kapitel_A.tex @@ -909,7 +909,7 @@ has been subsituted with $\hat{r_i}$. \dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{t_u}} = c \end{array} \end{equation} -Then by substituting the equation terms from \eqref{eq:MultitaylorDeriv}, \eqref{eq:rhoSatsNewFun} and \eqref{eq:rhoSatsNewFunApprox} +This is followed by substituting the equation terms from \eqref{eq:MultitaylorDeriv}, \eqref{eq:rhoSatsNewFun} and \eqref{eq:rhoSatsNewFunApprox} into \eqref{eq:MultitaylorFour}, the resulting equation is given in \eqref{eq:MultitaylorDerivAfter}. \begin{equation} \label{eq:MultitaylorDerivAfter} @@ -934,15 +934,15 @@ At this step, by solving equation \eqref{eq:MultitaylorFour}, the linearization \end{equation} By rearanging the equation \eqref{eq:MultitaylorDerivAfter} one derives equation \eqref{eq:MultitaylorDerivAfterRearange}. And then by substituting the terms in \eqref{eq:SubsTerms1} and \eqref{eq:SubsTerms2} into \eqref{eq:MultitaylorDerivAfterRearange}, -the equation resembles the one given in \eqref{eq:userPosition}. +the equation resembles the equation in \eqref{eq:userPosition}. \begin{equation} \label{eq:userPosition} \Delta\rho_i = \alpha_{xi}\Delta x_u + \alpha_{yi}\Delta y_u + \alpha_{zi}\Delta z_u - c\Delta t_u \end{equation} There are four unknowns, $\Delta x_u$, $\Delta y_u$, $\Delta z_u$ and $\Delta t_u$, in equation \eqref{eq:userPosition}. -By solving this set of linear equations, which shall result in finding $\Delta x_u$, $\Delta y_u$, $\Delta z_u$ and $\Delta t_u$, -the GPS receiver position $(x_u, y_u, z_u)$ and clock offset $t_u$ is computed by replacing the -same into equations in \eqref{eq:userCoordinates}. Equation \eqref{eq:userPosition} can be rewritten for four satellites +By solving this set of linear equations, which shall result in finding of $\Delta x_u$, $\Delta y_u$, $\Delta z_u$ and $\Delta t_u$, +the GPS receiver position is computed. The GPS receiver position $(x_u, y_u, z_u)$ and clock offset $t_u$ are obtained by substituting +them into equations in \eqref{eq:userCoordinates}. Equation \eqref{eq:userPosition} can be rewritten for four satellites in the matrix form as in \eqref{eq:userPositionMatrix}. \begin{equation} \label{eq:userPositionMatrix} @@ -974,8 +974,9 @@ in the matrix form as in \eqref{eq:userPositionMatrix}. -\Delta ct_u \end{bmatrix} \end{equation} -Finally, by multiplying both left sides\footnote{Matrix multiplication is not communitative, $\mathbf{AB\neq BA}$.} of the equation \eqref{eq:userPositionMatrix} -with the inverse term of $\boldsymbol{\alpha}$, it yields the result of the unknown terms, as given in equation \eqref{eq:userPositionMatrixFinal}. +Finally, by multiplying both left sides\footnote{Matrix multiplication is not communitative, $\mathbf{AB\neq BA}$.} +of the equation \eqref{eq:userPositionMatrix} with the inverse term of $\boldsymbol{\alpha}$, it yields the result +of the unknown terms, as given in equation \eqref{eq:userPositionMatrixFinal}. \begin{equation} \label{eq:userPositionMatrixInverseMult} \boldsymbol{\alpha}^{-1}\Delta\boldsymbol{\rho} = \boldsymbol{\alpha}^{-1}\boldsymbol{\alpha} \Delta \boldsymbol{x} @@ -989,7 +990,7 @@ to the just derived position values, that is, $\hat{x_u}=x_u$, $\hat{y_u}=y_u$, $\hat{t_u}=t_u$. This process is repeated until the approximated positions converge to their final values. It is not necessarily required that the initial positions are very accurate and the results are usually obtained by 4-5 itterations \citep{pseudorangeError}. -Risks exist that the solutions shall still be corrupted but there are different error avoiding +Risks exist that the solution may be still be corrupted but there are different error avoiding mechanisms to solve these problems, like minimizing the error contribution using more than four satellite measurements \citep{pseudorangeError} \citep[Chapter 7]{understandGPS}. -- cgit v1.2.3-55-g7522