From adc881ec84a2819b50a4961bfceaf4c4967a5c67 Mon Sep 17 00:00:00 2001 From: Martin Dieblich Date: Wed, 22 Sep 2010 17:12:40 +0000 Subject: [PATCH] Added documentation for the pprz_geodetic files, which are stored in the directory /sw/airborne/math --- doc/pprz_algebra/headfile.pdf | Bin 472080 -> 445746 bytes doc/pprz_algebra/headfile.tex | 4 +- doc/pprz_geodetic/Makefile | 8 + doc/pprz_geodetic/ecef.tex | 40 +++++ doc/pprz_geodetic/headfile.bib | 16 ++ doc/pprz_geodetic/headfile.pdf | Bin 0 -> 271443 bytes doc/pprz_geodetic/headfile.tex | 106 +++++++++++++ doc/pprz_geodetic/image-licence | 4 + doc/pprz_geodetic/images/ECEF.png | Bin 0 -> 30078 bytes doc/pprz_geodetic/introduction.tex | 4 + doc/pprz_geodetic/lla.tex | 46 ++++++ doc/pprz_geodetic/ltp.tex | 17 ++ doc/pprz_geodetic/ned_enu.tex | 53 +++++++ doc/pprz_geodetic/rates.tex | 145 ++++++++++++++++++ .../transformations/ecef2lla.tex | 29 ++++ .../transformations/ecef2ltp.tex | 15 ++ .../transformations/ecef2ned_enu.tex | 23 +++ .../transformations/lla2ecef.tex | 19 +++ .../transformations/lla2ned_enu.tex | 3 + .../transformations/ned_enu2ecef.tex | 20 +++ 20 files changed, 550 insertions(+), 2 deletions(-) create mode 100644 doc/pprz_geodetic/Makefile create mode 100644 doc/pprz_geodetic/ecef.tex create mode 100644 doc/pprz_geodetic/headfile.bib create mode 100644 doc/pprz_geodetic/headfile.pdf create mode 100644 doc/pprz_geodetic/headfile.tex create mode 100644 doc/pprz_geodetic/image-licence create mode 100644 doc/pprz_geodetic/images/ECEF.png create mode 100644 doc/pprz_geodetic/introduction.tex create mode 100644 doc/pprz_geodetic/lla.tex create mode 100644 doc/pprz_geodetic/ltp.tex create mode 100644 doc/pprz_geodetic/ned_enu.tex create mode 100644 doc/pprz_geodetic/rates.tex create mode 100644 doc/pprz_geodetic/transformations/ecef2lla.tex create mode 100644 doc/pprz_geodetic/transformations/ecef2ltp.tex create mode 100644 doc/pprz_geodetic/transformations/ecef2ned_enu.tex create mode 100644 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a/doc/pprz_algebra/headfile.tex b/doc/pprz_algebra/headfile.tex index 09cbd77280..8089e9a10e 100644 --- a/doc/pprz_algebra/headfile.tex +++ b/doc/pprz_algebra/headfile.tex @@ -92,7 +92,7 @@ \include{rates} \include{quaternion} \bibliographystyle{plain} -\bibliography{pprz} +\bibliography{headfile} \include{appendix} % -\end{document} \ No newline at end of file +\end{document} diff --git a/doc/pprz_geodetic/Makefile b/doc/pprz_geodetic/Makefile new file mode 100644 index 0000000000..47828837b5 --- /dev/null +++ b/doc/pprz_geodetic/Makefile @@ -0,0 +1,8 @@ +doc_pprz_algebra.pdf: headfile.tex + pdflatex $< +bib: + bibtex headfile + +clean: + rm -f *~ *.aux *.bbl *.blg *.log *.out *.toc *.dvi *.ps + find . -name '*~' -exec rm -f {} \; diff --git a/doc/pprz_geodetic/ecef.tex b/doc/pprz_geodetic/ecef.tex new file mode 100644 index 0000000000..b802753f27 --- /dev/null +++ b/doc/pprz_geodetic/ecef.tex @@ -0,0 +1,40 @@ +\section{ECEF} +Earth-Centered-Earth-Fixed coordinates belong to the easiest coordinates to compute, because they have a fixed frame and they don't have to regard the non-spherical shape of the earth. Though they are not intuitiv. \\ +\begin{figure}[h!] + \centering + \includegraphics[width=10cm]{ECEF} + \caption{ECEF coordinates} + \label{ECEF coordinates} +\end{figure} + +ECEF coordinates are defined using +\begin{equation} +p_{ECEF} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} +\end{equation} +available for the following simple types +\begin{itemize} +\item int32\_t - \texttt{EcefCoor\_i} +\item float - \texttt{EcefCoor\_f} +\item double - \texttt{EcefCoor\_d} +\end{itemize} + + + + +\subsection{Transformations from ECEF} +\subsubsection*{to LTP} +\input{transformations/ecef2ltp} + +\subsubsection*{to LLA} +\input{transformations/ecef2lla} + +\subsubsection*{to NED/ENU} +\input{transformations/ecef2ned_enu} + + +\subsection{Transformations to ECEF} +\subsubsection*{from LLA} +\input{transformations/lla2ecef} + +\subsubsection*{from NED/ENU} +\input{transformations/ned_enu2ecef} diff --git a/doc/pprz_geodetic/headfile.bib b/doc/pprz_geodetic/headfile.bib new file mode 100644 index 0000000000..7f6c948a42 --- /dev/null +++ b/doc/pprz_geodetic/headfile.bib @@ -0,0 +1,16 @@ + @misc{ wiki:1, + author = "Wikipedia", + title = "Geodetic system --- Wikipedia{,} The Free Encyclopedia", + year = "2010", + url = "\url{http://en.wikipedia.org/w/index.php?title=Geodetic_system&oldid=382672935}", + note = "[Online; accessed 22-September-2010]" + } +@book{Wendel__2007, + author = {Wendel, Jan}, + citeulike-article-id = {7547540}, + posted-at = {2010-07-28 08:23:54}, + priority = {2}, + publisher = {Oldenbourg}, + title = {Integrierte Navigationssysteme. 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+\usepackage{url} + +% For drawings +\usepackage{pgf} +\usepackage{tikz} +\usetikzlibrary{arrows,automata} +\usepackage{color} + +% For quick and easy figures +\newcommand{\pic}[3]{ +\begin{figure}[h]\begin{center}\includegraphics[width=#2]{#1.png} + \caption{#3} + \label{#1} +\end{center}\end{figure} +} + +%\newcommand{\inHfile}[2]{ +% Function \texttt{#1} in File \texttt{#2.h} \\ +%} +%\newcommand{\inCfile}[2]{ +% Function \texttt{#1} in File \texttt{#2.c} \\ +%} + +\newcommand{\inHfile}[2]{Function \texttt{#1}\\ +\indent in File \texttt{#2.h} \\} + +\newcommand{\inCfile}[2]{Function \texttt{#1}\\ +\indent in File \texttt{#2.c} \\} + +%\numberwithin{equation}{section} +\newcommand{\vect}[1]{\ensuremath{\overrightarrow{#1}}} %% How to mark Vectors +\newcommand{\eu}[1]{\ensuremath{#1^{\phi}}} %% how to mark euler angles +\newcommand{\ra}[1]{\ensuremath{\omega_{#1}}} %% how to mark rates +\newcommand{\ew}[1]{\;. \! \!#1} %% how to mark element-wise operations +\newcommand{\mat}[1]{\ensuremath{\mathbf{#1}}} %% how to mark Matrices +\newcommand{\eye}[0]{\mat{I}} %% Identity matrix +\newcommand{\quat}[1]{\ensuremath{q_{#1}}} %% how to mark Quaternions +\newcommand{\transp}[1]{\ensuremath{#1^{T}}} +\newcommand{\est}[1]{\ensuremath{\hat{#1}}} +\newcommand{\err}[1]{\ensuremath{\tilde{#1}}} +\newcommand{\meas}[1]{\ensuremath{\tilde{#1}}} +\newcommand{\linpt}[1]{\ensuremath{\overline{#1}}} +\newcommand{\norm}[1]{\ensuremath{|{#1}|}} +\newcommand{\quatprod}[0]{\ensuremath{\bullet}} +\newcommand{\ddt}[2]{\ensuremath{#1^{(#2)}}} +\newcommand{\deriv}[2]{\ensuremath{{#1}^{(#2)}}} +\newcommand{\inv}[1]{\ensuremath{#1}^{-1}} +\newcommand{\comp}[1]{\ensuremath{#1}^*} +\newcommand{\atan}[1]{\ensuremath{\text{atan}\left({#1}\right)}} +\newcommand{\sign}[1]{\ensuremath{\text{sign}\left({#1}\right)}} +\newcommand{\cross}{\ensuremath{\times}} + +\newcommand{\division}[0]{\ensuremath{\div}} +\newcommand{\multiplication}[0]{\ensuremath{\cdot}} + + +%% Formatting in the right color for euler angles +\definecolor{rollcolor}{rgb}{0,0,1} +\definecolor{pitchcolor}{rgb}{0,0.5,0} +\definecolor{yawcolor}{rgb}{1,0,0} +\newcommand{\Rollc}[1]{\color{rollcolor}#1\color{black}{}} +\newcommand{\Pitchc}[1]{\color{pitchcolor}#1\color{black}{}} +\newcommand{\Yawc}[1]{\color{yawcolor}#1\color{black}{}} +\newcommand{\Roll}[0]{\ensuremath{\Rollc \phi}} +\newcommand{\Pitch}[0]{\ensuremath{\Pitchc \theta }} +\newcommand{\Yaw}[0]{\ensuremath{\Yawc \psi}} +\newcommand{\lon}[0]{\ensuremath{\lambda}} +\newcommand{\lat}[0]{\ensuremath{\varphi}} + + +\newcommand{\mynote}[1]{\begin{flushright}\fbox{Martin: ``\textit{#1}''}\end{flushright}} +%\newcommand{\mynote}[1]{} + +\graphicspath{{./images/},{tmp/}} + +\title{Documentation for pprz_algebra} +\author{Martin Dieblich} + +\begin{document} + +%\maketitle +\include{introduction} +\tableofcontents +\include{ecef} +\include{lla} +\include{ltp} +\include{ned_enu} +\bibliographystyle{acm} +\bibliography{headfile} +% +\end{document} \ No newline at end of file diff --git a/doc/pprz_geodetic/image-licence b/doc/pprz_geodetic/image-licence new file mode 100644 index 0000000000..db0e4f4aad --- /dev/null +++ b/doc/pprz_geodetic/image-licence @@ -0,0 +1,4 @@ +FILE: ECEF.png +URI: http://en.wikipedia.org/wiki/File:ECEF.png +LICENSE: PD (public domain) +AUTHOR: Kr7cmw0l ( http://commons.wikimedia.org/wiki/User:Kr7cmw0l ) diff --git a/doc/pprz_geodetic/images/ECEF.png b/doc/pprz_geodetic/images/ECEF.png new file mode 100644 index 0000000000000000000000000000000000000000..4acd9708febeb585eeba769e2ed78eb953aae799 GIT binary patch literal 30078 zcmcHg^;?wR_XmtJHvdH7(gG?LXAE={}juH+|O#<lhB ziYs-3SppqllSh5bvh09a@lfo2JHq*ge~3Ph-&|i`U3|F>Q^Wz_C>$s&4n!4-59$4X zeo&yEi7x#g`CrSGRU_X2e+Srab`)xCRixt0fBpY8$iz_K{{Px5GD38*^?oD)Y&ekr z=J}nC15{nW!5xFs{rfTtWBq@%fu|Do;>CHOT!aarC_ZD|{B=2Kpj)VkW|KIwwp=vV+qb_y+88j1@2T-M79N59x)5C=` za>$ADD-_3Pa9g3!%$3lMACIq-J`KPuIk=T_DMO{RQVt(!aVUXT 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\ra v \leq \ra{v_{max}}$ Bounding} +Ensures that +\begin{equation} +\ra {v_{min}} \leq \ra v \leq \ra{v_{max}} \Leftrightarrow \begin{pmatrix} p_{min} \\ q_{min} \\ r_{min} \end{pmatrix} \leq \begin{pmatrix} p \\ q \\ r \end{pmatrix} \leq \begin{pmatrix} p_{max} \\ q_{max} \\ r_{max} \end{pmatrix} +\end{equation} +The upper border \textit{max} has a higher priority than \textit{min}. So, if $ min > max$ and a value of $ \ra v $ is between those, the value is set to \textit{max}. \\ +\inHfile{RATES\_BOUND\_BOX(v, v\_min, v\_max}{pprz\_algebra} +\mynote{Not very consequent with the priority} \ No newline at end of file diff --git a/doc/pprz_geodetic/transformations/ecef2lla.tex b/doc/pprz_geodetic/transformations/ecef2lla.tex new file mode 100644 index 0000000000..3a05656eb2 --- /dev/null +++ b/doc/pprz_geodetic/transformations/ecef2lla.tex @@ -0,0 +1,29 @@ +Generating LLA coordinates is made using the following calculations. These refer to \cite{wiki:1} or \cite{Wendel__2007}(pages 31-33). + +\begin{align} +a &= 6,378,137 \\ +f &= \frac{1}{298.257223563} +\end{align} +\begin{align} +b &= a \multiplication (1-f) \\ +e^2 &= \sqrt{2f-f^2} \\ +e' &= e \frac{a}{b} \\ +E^2 &= a^2-b^2 \\ +r &= \sqrt{ x^2 + y^2} \\ +F &= 54 b^2 z^2 \\ +G &= r^2 + (1-e^2)z^2-e^2E^2 \\ +c &= \frac{e^4Fr^2}{G^3} \\ +s &= \sqrt[3]{1+c+\sqrt{c^2+2c}} \\ +P &= \frac{F}{3\left(s+\tfrac 1 s + 1\right)^2 G^2} \\ +Q &= \sqrt{1+2e^4P} \\ +r_0 &= -\frac{Pe^2r}{1+Q} + \sqrt{\tfrac 1 2 a^2 \left(1 + \tfrac 1 Q \right) - \frac{P(1-e^2)z^2}{Q(1+Q)}-\tfrac 1 2 P r^2} \\ +U &= \sqrt{(r-e^2r_0)^2+z^2} \\ +V &= \sqrt{(r-e^2r_0)^2+(1-e^2)z^2} \\ +z_0 &= \frac{b^2z}{aV} \\ +\lat &= \arctan \left( \frac{z+(e')^2z_0} r \right) \\ +\lon &= atan2(y,x) \\ +h &= U \left( \frac{b^2}{aV} - 1 \right) +\end{align} +\inCfile{lla\_of\_ecef\_i(LlaCoor\_i* out, EcefCoor\_i* in)}{pprz\_geodetic\_int} +\inCfile{lla\_of\_ecef\_f(LlaCoor\_f* out, EcefCoor\_f* in)}{pprz\_geodetic\_float} +\inCfile{lla\_of\_ecef\_d(LlaCoor\_d* lla, EcefCoor\_d* ecef)}{pprz\_geodetic\_double} \ No newline at end of file diff --git a/doc/pprz_geodetic/transformations/ecef2ltp.tex b/doc/pprz_geodetic/transformations/ecef2ltp.tex new file mode 100644 index 0000000000..fc696a3236 --- /dev/null +++ b/doc/pprz_geodetic/transformations/ecef2ltp.tex @@ -0,0 +1,15 @@ +Gets the LLA-coordinates and a transformation matrix from ECEF to ENU out of ECEF coordinates. + +First, the LLA coordinates (WGS-84) are computed from the ECEF coordinates. + +With the LLA-coordinates it is possible to construct a rotational matrix. +\begin{equation} +\mat R_{ecef2enu} = \begin{pmatrix} +-\sin \lon & \cos \lon & 0 \\ +-\sin \lat \cos \lon & -\sin \lat \sin \lon & \cos \lat \\ +\cos \lat \cos \lon & \cos \lat \sin \lon & \sin \lat +\end{pmatrix} +\end{equation} +\inCfile{ltp\_def\_from\_ecef\_i(LtpDef\_i* def, EcefCoor\_i* ecef)}{pprz\_geodetic\_int} +\inCfile{ltp\_def\_from\_ecef\_f(LtpDef\_f* def, EcefCoor\_f* ecef)}{pprz\_geodetic\_float} +\inCfile{ltp\_def\_from\_ecef\_d(LtpDef\_d* def, EcefCoor\_d* ecef)}{pprz\_geodetic\_double} \ No newline at end of file diff --git a/doc/pprz_geodetic/transformations/ecef2ned_enu.tex b/doc/pprz_geodetic/transformations/ecef2ned_enu.tex new file mode 100644 index 0000000000..8551189a5d --- /dev/null +++ b/doc/pprz_geodetic/transformations/ecef2ned_enu.tex @@ -0,0 +1,23 @@ +With a know transformation matrix (see section \ref{LTP}) it is quite easy to rotate a vector into the ENU frame: +\begin{equation} +\vect v_{ENU} = \mat R_{ecef2enu} \vect v_{ECEF} +\end{equation} +For a transformation into the NED-frame you have to do an additional ENU/NED-transformation. + +\inCfile{enu\_of\_ecef\_vect\_i(EnuCoor\_i* enu, LtpDef\_i* def, EcefCoor\_i* ecef)}{pprz\_geodetic\_int} +\inCfile{ned\_of\_ecef\_vect\_i(NedCoor\_i* ned, LtpDef\_i* def, EcefCoor\_i* ecef)}{pprz\_geodetic\_int} +\inCfile{enu\_of\_ecef\_vect\_f(EnuCoor\_f* enu, LtpDef\_f* def, EcefCoor\_f* ecef)}{pprz\_geodetic\_float} +\inCfile{ned\_of\_ecef\_vect\_f(NedCoor\_f* ned, LtpDef\_f* def, EcefCoor\_f* ecef)}{pprz\_geodetic\_float} +\inCfile{enu\_of\_ecef\_vect\_d(EnuCoor\_d* enu, LtpDef\_d* def, EcefCoor\_d* ecef)}{pprz\_geodetic\_double} +\inCfile{ned\_of\_ecef\_vect\_d(NedCoor\_d* ned, LtpDef\_d* def, EcefCoor\_d* ecef)}{pprz\_geodetic\_double} + +The transformation of a point is very similiar. Instead of a point you use a difference vector between the desired point $p_d$ and the center of the local tangent plane $p_0$. +\begin{equation} +\vect v_{ECEF} = p_d - p_0 +\end{equation} +\inCfile{enu\_of\_ecef\_point\_i(EnuCoor\_i* enu, LtpDef\_i* def, EcefCoor\_i* ecef)}{pprz\_geodetic\_int} +\inCfile{ned\_of\_ecef\_point\_i(NedCoor\_i* ned, LtpDef\_i* def, EcefCoor\_i* ecef)}{pprz\_geodetic\_int} +\inCfile{enu\_of\_ecef\_point\_f(EnuCoor\_f* enu, LtpDef\_f* def, EcefCoor\_f* ecef)}{pprz\_geodetic\_float} +\inCfile{ned\_of\_ecef\_point\_f(NedCoor\_f* ned, LtpDef\_f* def, EcefCoor\_f* ecef)}{pprz\_geodetic\_float} +\inCfile{enu\_of\_ecef\_point\_d(EnuCoor\_d* enu, LtpDef\_d* def, EcefCoor\_d* ecef)}{pprz\_geodetic\_double} +\inCfile{ned\_of\_ecef\_point\_d(NedCoor\_d* ned, LtpDef\_d* def, EcefCoor\_d* ecef)}{pprz\_geodetic\_double} \ No newline at end of file diff --git a/doc/pprz_geodetic/transformations/lla2ecef.tex b/doc/pprz_geodetic/transformations/lla2ecef.tex new file mode 100644 index 0000000000..872baa0277 --- /dev/null +++ b/doc/pprz_geodetic/transformations/lla2ecef.tex @@ -0,0 +1,19 @@ +Calculating the ECEF coordinates out of LLA coordinates is a slightly easier task than the other way round. The following way refers to \cite{wiki:1}. With the known constants +\begin{align} +a &= 6,378,137 \\ +f &= \frac{1}{298.257223563} \\ +e^2 &= \sqrt{2f-f^2} +\end{align} +the value +\begin{equation} +\chi = \sqrt{1-e^2\sin^2 \lat} +\end{equation} +can be precomputed and used in +\begin{align} +x &= \left(\tfrac a \chi + h \right) \cos \lat \cos \lon \\ +y &= \left(\tfrac a \chi + h \right) \cos \lat \sin \lon \\ +z &= \left(\tfrac a \chi (1-e^2) + h \right) \sin \lat +\end{align} +\inCfile{ecef\_of\_lla\_i(EcefCoor\_i* out, LlaCoor\_i* in)}{pprz\_geodetic\_int} +\inCfile{ecef\_of\_lla\_f(EcefCoor\_f* out, LlaCoor\_f* in)}{pprz\_geodetic\_float} +\inCfile{ecef\_of\_lla\_d(EcefCoor\_d* ecef, LlaCoor\_d* lla)}{pprz\_geodetic\_double} \ No newline at end of file diff --git a/doc/pprz_geodetic/transformations/lla2ned_enu.tex b/doc/pprz_geodetic/transformations/lla2ned_enu.tex new file mode 100644 index 0000000000..65de832a77 --- /dev/null +++ b/doc/pprz_geodetic/transformations/lla2ned_enu.tex @@ -0,0 +1,3 @@ +This functions transforms a point from the LLA coordinates in the ECEF-frame and then into the NED-/ENU-frame. +\inCfile{enu\_of\_lla\_point\_i(EnuCoor\_i* enu, LtpDef\_i* def, LlaCoor\_i* lla)}{pprz\_geodetic\_int} +\inCfile{ned\_of\_lla\_point\_i(NedCoor\_i* ned, LtpDef\_i* def, LlaCoor\_i* lla)}{pprz\_geodetic\_int} \ No newline at end of file diff --git a/doc/pprz_geodetic/transformations/ned_enu2ecef.tex b/doc/pprz_geodetic/transformations/ned_enu2ecef.tex new file mode 100644 index 0000000000..789089f74f --- /dev/null +++ b/doc/pprz_geodetic/transformations/ned_enu2ecef.tex @@ -0,0 +1,20 @@ +With a know transformation matrix (see section \ref{LTP}) it is quite easy to rotate a vector from the ENU-frame to the ECEF-frame. +Since +\begin{equation} +\mat R_{enu2ecef} = \inv{\mat R_{ecef2enu}} = \transp{\mat R_{ecef2enu}} +\end{equation} +a transformation is done as follows +\begin{equation} +\vect v_{ECEF} = \mat R_{enu2ecef} \vect v_{ENU} +\end{equation} +For a transformation from the NED-frame you have to do an additional ENU/NED-transformation before. + +\inCfile{ecef\_of\_enu\_vect\_d(EcefCoor\_d* ecef, LtpDef\_d* def, EnuCoor\_d* enu)}{pprz\_geodetic\_double} +\inCfile{ecef\_of\_ned\_vect\_d(EcefCoor\_d* ecef, LtpDef\_d* def, NedCoor\_d* ned)}{pprz\_geodetic\_double} + +The transformation of a point is very similiar. After transforming into the ECEF-frame you add the position of the center $p_0$ to the result. +\begin{equation} +\vect p_{ECEF} = p + p_0 +\end{equation} +\inCfile{ecef\_of\_enu\_point\_d(EcefCoor\_d* ecef, LtpDef\_d* def, EnuCoor\_d* enu)}{pprz\_geodetic\_double} +\inCfile{ecef\_of\_ned\_point\_d(EcefCoor\_d* ecef, LtpDef\_d* def, NedCoor\_d* ned)}{pprz\_geodetic\_double} \ No newline at end of file