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[paparazzi-commits] [6227]
From: |
Martin Dieblich |
Subject: |
[paparazzi-commits] [6227] |
Date: |
Mon, 25 Oct 2010 11:32:12 +0000 |
Revision: 6227
http://svn.sv.gnu.org/viewvc/?view=rev&root=paparazzi&revision=6227
Author: mdieblich
Date: 2010-10-25 11:32:12 +0000 (Mon, 25 Oct 2010)
Log Message:
-----------
Modified Paths:
--------------
paparazzi3/trunk/sw/airborne/fms/libeknav/Makefile
paparazzi3/trunk/sw/airborne/fms/libeknav/doc/content.tex
paparazzi3/trunk/sw/airborne/fms/libeknav/doc/headfile.pdf
paparazzi3/trunk/sw/airborne/fms/libeknav/doc/headfile.tex
paparazzi3/trunk/sw/airborne/fms/libeknav/estimate_attitude.c
paparazzi3/trunk/sw/airborne/fms/libeknav/estimate_attitude.h
paparazzi3/trunk/sw/airborne/fms/libeknav/ins_qkf.hpp
paparazzi3/trunk/sw/airborne/fms/libeknav/ins_qkf_observe_gps_p.cpp
paparazzi3/trunk/sw/airborne/fms/libeknav/ins_qkf_observe_gps_pvt.cpp
paparazzi3/trunk/sw/airborne/fms/libeknav/libeknav_from_log.cpp
paparazzi3/trunk/sw/airborne/fms/libeknav/libeknav_from_log.hpp
paparazzi3/trunk/sw/airborne/fms/libeknav/paparazzi_eigen_conversion.h
paparazzi3/trunk/sw/airborne/fms/libeknav/raw_log_to_ascii.c
paparazzi3/trunk/sw/airborne/fms/libeknav/test_libeknav_4.cpp
paparazzi3/trunk/sw/airborne/fms/libeknav/test_libeknav_4.hpp
Modified: paparazzi3/trunk/sw/airborne/fms/libeknav/Makefile
===================================================================
--- paparazzi3/trunk/sw/airborne/fms/libeknav/Makefile 2010-10-25 08:58:08 UTC
(rev 6226)
+++ paparazzi3/trunk/sw/airborne/fms/libeknav/Makefile 2010-10-25 11:32:12 UTC
(rev 6227)
@@ -20,7 +20,7 @@
run_filter_on_log: ./libeknav_from_log.cpp $(LIBEKNAV_SRCS)
../../math/pprz_geodetic_double.c ../../math/pprz_geodetic_float.c
- g++ -I/usr/include/eigen2 -I../.. -I../../../include
-I../../../../var/FY -DOVERO_LINK_MSG_UP=AutopilotMessageVIUp
-DOVERO_LINK_MSG_DOWN=AutopilotMessageVIDown -o $@ $^
+ g++ -I/usr/include/eigen2 -I../.. -I../../../include
-I../../../../var/FY -DOVERO_LINK_MSG_UP=AutopilotMessageVIUp
-DOVERO_LINK_MSG_DOWN=AutopilotMessageVIDown -DEKNAV_FROM_LOG_DEBUG -o $@ $^
clean:
-rm -f *.o *~ *.d
Modified: paparazzi3/trunk/sw/airborne/fms/libeknav/doc/content.tex
===================================================================
--- paparazzi3/trunk/sw/airborne/fms/libeknav/doc/content.tex 2010-10-25
08:58:08 UTC (rev 6226)
+++ paparazzi3/trunk/sw/airborne/fms/libeknav/doc/content.tex 2010-10-25
11:32:12 UTC (rev 6227)
@@ -41,6 +41,7 @@
\section{Initialisation}
\subsection{What about the standard deviation?}
+Pay attention to the following equations! Many things are just guesses and no
really mathematically proofen. So better think twice before just using my
assumptions.
\begin{figure}[h!]\begin{center}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm,
semithick]
@@ -63,14 +64,16 @@
\caption{Propagation of uncertainty}
\label{Propagation of uncertainty}
\end{center}\end{figure}
+\subsubsection*{Measurement}
First of all, every sensor (accelerometers $\vect a$ and magnetometers $\vect
m$) has gaussian noise, that can be expressed as an additive error:
\begin{equation}
\vect a + \vect{\sigma_a} \quad \quad \vect m + \vect{\sigma_m}
\end{equation}
It can be asssumed that the error follows a standard deviation (has zero mean
and is time-invariant).
+\subsubsection*{attitude profile matrix}
The attitude profile matrix $ \mat B $ is the sum of the measurements with
specific weights.
\begin{equation}\label{attitude profile matrix}
-\mat B = \sum_{k=1}^n w_k \cdot \vect{W}_k \cdot \transp{\vect{V}_k} = w_a
\sum_{k=1}^{n_a} \frac{\vect a_k}{\norm{a_k}} \cdot \transp{\vect{g}} + w_m
\sum_{k=1}^{n_m} \frac{\vect m_k}{\norm{m_k}} \cdot \transp{\vect{h}}
+\mat B = \sum_{k=1}^n w_k \cdot \vect{W}_k \cdot \transp{\vect{V}_k} = w_a
\sum_{k=1}^{n_a} \vect a_k \cdot \transp{\vect{g}} + w_m \sum_{k=1}^{n_m}
\vect m_k \cdot \transp{\vect{h}}
\end{equation}
$n$ is the number of measurements, $w_k$ is the specific weight of a
measurement, $\vect{W}_k$ the measured vector and $\vect{V}_k$ the reference
direction, which belongs to the measured direction. Therefore $n_a$ is the
number of acceleration measurements, $w_a$ is the (constant) weight of the
acceleration measurements, $\vect{a}_k$ is a single acceleration observation
and $\vect{g}$ is the gravity. $\vect{a}_k$ becomes normed. Similar for the
magnetometer weight $w_m$, measurement $\vect {m}_k$, the magnetic field $\vect
h$ and the amount of magnetometer measurements $n_m$. See the next section how
the weight should be choosen.
@@ -78,7 +81,7 @@
\begin{equation}
\mat{\sigma_B} = \frac{n_a}{f_a} \frac{1}{\norm g_2}
\vect{\sigma_a}\transp{\vect{g}} + \frac{n_m}{f_m} \frac{1}{\norm h_2}
\vect{\sigma_m}\transp{\vect{m}}
\end{equation}
-
+\subsubsection*{``K''-matrix}
The error for the ``K''-matrix is easy to get by inserting $ \mat B + \mat
{\sigma_B} $ into
\begin{equation}
\mat K = \begin{bmatrix}
@@ -86,9 +89,68 @@
\vect Z & \mat B + \transp{\mat B} - trace(\mat B) \mat I
\end{bmatrix}
\end{equation}
+\begin{equation}
+\mat{\sigma_K} = \begin{bmatrix}
+trace(\mat{\sigma_B}) & \transp{\vect \sigma_Z} \\
+\vect \sigma_Z & \mat{\sigma_B} + \transp{\mat{\sigma_B}} -
trace(\mat{\sigma_B}) \mat I
+\end{bmatrix}
+\end{equation}
+\subsubsection*{Eigenvector}
+The dominant eigenvector is computed with the power iteration:
+\begin{equation}
+\vect x_{k+1} = \frac{\mat K \vect x_k}{\norm{\mat K \vect x_k}_{max}}
+\end{equation}
+With $ \mat K \rightarrow \mat K + \mat {\sigma_K} $:
+\begin{equation}
+\vect x_{k+1} + \vect\sigma_{x_{k+1}} = \frac{(\mat K + \mat {\sigma_K}) \vect
x_k}{\norm{(\mat K + \mat {\sigma_K}) \vect x_k}_{max}} = \frac{\mat K \vect
x_k}{\norm{(\mat K + \mat {\sigma_K}) \vect x_k}_{max}} + \frac{\mat {\sigma_K}
\vect x_k}{\norm{(\mat K + \mat {\sigma_K}) \vect x_k}_{max}}
+\end{equation}
+If we assume that
+\begin{equation}
+\norm{(\mat K + \mat {\sigma_K}) \vect x_k}_{max} \approx \norm{\mat K \vect
x_k}_{max}
+\end{equation}
+we'll get:
+\begin{equation}
+\vect\sigma_{x_{k+1}} = \frac{\mat {\sigma_K} \vect x_k}{\norm{\mat K\vect
x_k}_{max}}
+\end{equation}
+This means that our final error at the end of the iteration can be computed
using the vector from the step before:
+\begin{equation}
+\vect\sigma_{x_n} = \frac{\mat {\sigma_K} \vect x_{n-1}}{\norm{\mat K\vect
x_{n-1}}_{max}}
+\end{equation}
+We should keep in mind, that we get an additional error because the the power
iteration does not stop, when it's close to the eigenvector. It stops if two
iteration steps $\vect x_k $ and $\vect x_{k+1} $ are close to each other. To
avoid getting an additional problem with that we should choose the canceling
condition of the iteration $\delta = \norm{\vect x_k-\vect x_{k+1}}_{max} $
much smaller than $\norm{\mat{\sigma_K}}_{max} $.
+And, of course:
+\begin{equation}
+\vect \sigma_x = \quat{\sigma}
+\end{equation}
+\subsubsection*{Euler angles}
+Our current expression is something like $ \quat{true} = \quat{false} +
\sigma_{\quat{}} $. But it would be helpful to express the difference as a
multiplication:
+\begin{align}
+\quat{false} + \sigma_{\quat{}} &=
\quat{false2true} \quatprod \quat{false} \\
+(\quat{} + \sigma_{\quat{}}) \quatprod \quat{false}^{-1} &= \quat{false2true}
\quatprod \quat{false} \quatprod \quat{false}^{-1} \\
+\quat{false2true} &= (\quat{false} + \sigma_{\quat{}}) \quatprod
\quat{false}^{-1} \\
+\quat{false2true} &= \mathbf{I} + \sigma_{\quat{}} \quatprod \quat{false}^{-1}
\\
+\quat{false2true} &= \mathbf{I} + \sigma_{\quat{}} \quatprod
\comp{\quat{false}} \label{equ. false2true}
+\end{align}
+Composition of small Euler angles\footnote{\emph{only} small Euler angles!} is
done by simply adding them. We can assume that the error of the quaternion is
small, so addition should be valid.
+\begin{equation}
+\quat{false2true} \quatprod \quat{false} \rightarrow \begin{pmatrix}\phi \\
\theta \\ \psi \end{pmatrix} + \begin{pmatrix}\sigma_{\phi} \\ \sigma_{\theta}
\\ \sigma_{\psi} \end{pmatrix} \\
+\quat{false2true} \rightarrow \begin{pmatrix}\sigma_{\phi} \\ \sigma_{\theta}
\\ \sigma_{\psi} \end{pmatrix}
+\end{equation}
+For small rotations, the imaginary part $\vect v$ of the quaternion becomes
Euler angles.
+\begin{equation}
+\vect v_{false} = \begin{pmatrix}\sigma_{\phi} \\ \sigma_{\theta} \\
\sigma_{\psi} \end{pmatrix} \label{equ. err euler}
+\end{equation}
+As a result, we don't care about the Identity rotation from equation
(\ref{equ. false2true}). Furthermore, with
+\begin{equation}
+\quat{false} = \begin{pmatrix} \quat 0 \\ \vect v_q \end{pmatrix} \quad and
\quad
+\sigma_q = \begin{pmatrix} \sigma_{q0} \\ \vect v_{\sigma} \end{pmatrix}
+\end{equation}
+we can rewrite equation (\ref{equ. err euler}) to
+\begin{equation}
+\begin{pmatrix}\sigma_{\phi} \\ \sigma_{\theta} \\ \sigma_{\psi} \end{pmatrix}
= \quat 0 \vect v_{\sigma} - \sigma_{q0} \vect v_q + \vect v_q \cross \vect
v_{\sigma}.
+\end{equation}
@@ -96,53 +158,52 @@
-
\subsection{choosing the best weight for the attitude profile matrix}
-If you replace the single measurements in equation (\ref{attitude profile
matrix}) with the real (and normed) measurements
+If you replace the single measurements in equation (\ref{attitude profile
matrix}) with the real measurements
\begin{equation}
-\frac{\vect a_k + \vect{\sigma_a}}{\norm{a_k}}_2 \quad \quad \frac{\vect m_k +
\vect{\sigma_m}}{\norm{m_k}}_2
+\vect a_k + \vect{\sigma_a} \quad \quad \vect m_k + \vect{\sigma_m}
\end{equation}
and assume that $\mat B$ has an error $\mat B +\mat{\sigma_B} $, you will get
\begin{equation}
-\mat B +\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} \frac{\vect a_k +
\vect{\sigma_a}}{\norm{a_k}_2} \cdot \transp{\vect{g}} + w_m \sum_{k=1}^{n_m}
\frac{\vect m_k + \vect{\sigma_m}}{\norm{m_k}_2} \cdot \transp{\vect{h}}
+\mat B +\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} (\vect a_k + \vect{\sigma_a})
\cdot \transp{\vect{g}} + w_m \sum_{k=1}^{n_m} (\vect m_k + \vect{\sigma_m})
\cdot \transp{\vect{h}}
\end{equation}
\begin{equation}
-\mat B +\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} \frac{\vect a_k}{\norm{a_k}}_2
\cdot \transp{\vect{g}} + \frac{\vect{\sigma_a}}{\norm{a_k}}_2 \cdot
\transp{\vect{g}} + w_m \sum_{k=1}^{n_m} \frac{\vect m_k}{\norm{m_k}}_2 \cdot
\transp{\vect{h}} + \frac{\vect{\sigma_m}}{\norm{m_k}}_2 \cdot \transp{\vect{h}}
+\mat B +\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} \vect a_k \cdot
\transp{\vect{g}} + \vect{\sigma_a} \cdot \transp{\vect{g}} + w_m
\sum_{k=1}^{n_m} \vect m_k \cdot \transp{\vect{h}} + \vect{\sigma_m} \cdot
\transp{\vect{h}}
\end{equation}
\begin{equation}
-\mat B +\mat{\sigma_B} = \underbrace{w_a \sum_{k=1}^{n_a} \frac{\vect
a_k}{\norm{a_k}}_2\cdot \transp{\vect{g}} + w_m \sum_{k=1}^{n_m} \frac{\vect
m_k}{\norm{m_k}}_2 \cdot \transp{\vect{h}}}_{\mat B} + w_a \sum_{k=1}^{n_a}
\frac{\vect{\sigma_a}}{\norm{a_k}}_2 \cdot \transp{\vect{g}} + w_m
\sum_{k=1}^{n_m} \frac{\vect{\sigma_m}}{\norm{m_k}}_2 \cdot \transp{\vect{h}}
+\mat B +\mat{\sigma_B} = \underbrace{w_a \sum_{k=1}^{n_a} \vect a_k \cdot
\transp{\vect{g}} + w_m \sum_{k=1}^{n_m} \vect m_k \cdot
\transp{\vect{h}}}_{\mat B} + w_a \sum_{k=1}^{n_a} \vect{\sigma_a} \cdot
\transp{\vect{g}} + w_m \sum_{k=1}^{n_m} \vect{\sigma_m} \cdot \transp{\vect{h}}
\end{equation}
\begin{equation}
-\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} \frac{\vect{\sigma_a}}{\norm{a_k}}_2
\cdot \transp{\vect{g}} + w_m \sum_{k=1}^{n_m}
\frac{\vect{\sigma_m}}{\norm{m_k}}_2 \cdot \transp{\vect{h}}
+\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} \vect{\sigma_a} \cdot \transp{\vect{g}}
+ w_m \sum_{k=1}^{n_m} \vect{\sigma_m} \cdot \transp{\vect{h}}
\end{equation}
-$\norm{a_k}_2$ and $\norm{m_k}_2$ shouldn't vary that much and can be assumed
as constant ($\norm{a}_2$ and $\norm{m}_2$). The equation reduces to:
+The sums are independent from their indices:
\begin{equation}
-\mat{\sigma_B} = w_a n_a \frac{\vect{\sigma_a}}{\norm{a}_2}\cdot
\transp{\vect{g}} + w_m n_m \frac{\vect{\sigma_m}}{\norm{m}_2} \cdot
\transp{\vect{h}}
+\mat{\sigma_B} = w_a n_a \vect{\sigma_a} \cdot \transp{\vect{g}} + w_m n_m
\vect{\sigma_m} \cdot \transp{\vect{h}}
\end{equation}
It would be nice, if it's possible to reduce this to a single value. To do
that, we need a matrix norm. In this case, I choosed the Frobenius Norm:
\begin{align}
-\norm{\mat{\sigma_B}}_{F} &= \norm{w_a n_a
\frac{\vect{\sigma_a}}{\norm{a}_2}\cdot \transp{\vect{g}} + w_m n_m
\frac{\vect{\sigma_m}}{\norm{m}_2} \cdot \transp{\vect{h}}}_{F} \\
-&\le \norm{w_a n_a \frac{\vect{\sigma_a}}{\norm{a}_2}\cdot
\transp{\vect{g}}}_{F} + \norm{w_m n_m \frac{\vect{\sigma_m}}{\norm{m}_2} \cdot
\transp{\vect{h}}}_{F} \\
-&= w_a n_a \frac{1}{\norm{a}_2}\cdot \norm{\vect{\sigma_a}
\transp{\vect{g}}}_{F} + w_m n_m \frac{1}{\norm{m}_2} \cdot
\norm{\vect{\sigma_m} \transp{\vect{h}}}_{F}
+\norm{\mat{\sigma_B}}_{F} &= \norm{w_a n_a \vect{\sigma_a} \cdot
\transp{\vect{g}} + w_m n_m \vect{\sigma_m} \cdot \transp{\vect{h}}}_{F}
\\
+&\le \norm{w_a n_a \vect{\sigma_a} \cdot \transp{\vect{g}}}_{F} + \norm{w_m
n_m \vect{\sigma_m} \cdot \transp{\vect{h}}}_{F} \\
+&= w_a n_a \norm{\vect{\sigma_a} \transp{\vect{g}}}_{F} + w_m n_m
\norm{\vect{\sigma_m} \transp{\vect{h}}}_{F}
\end{align}
It is straight-forward to proove that $ \norm{\vect a \transp{\vect b}}_F =
\norm a_2 \cdot \norm b_2 $
\begin{equation}
-\norm{\mat{\sigma_B}}_{F} \le w_a n_a \frac{\norm g_2}{\norm a_2}\cdot
\norm{\sigma_a}_2 + w_m n_m \frac{\norm h_2}{\norm m_2} \cdot \norm{\sigma_m}_2
+\norm{\mat{\sigma_B}}_{F} \le w_a n_a \norm g_2 \cdot \norm{\sigma_a}_2 + w_m
n_m \norm h_2 \cdot \norm{\sigma_m}_2
\end{equation}
As you can see, the uncertainty depends on the following parameters:
\begin{itemize}
\item The weight of a measurement $w_a$ and $w_m$.
\item The number of measurements $n_a$ and $n_m$.
-\item Something that I call a "measurement gain",
$\frac{\norm{a}_2}{\norm{g}_2}$ and $\frac{\norm{m}_2}{\norm{h}_2}$, since it's
the ratio between the true value and the measured value.
+\item The length/norm of the reference directions $\vect g$ and $\vect h$.
\item The maximum of the error $\sigma_a$ and $\sigma_m$.
\end{itemize}
-This is not what I want. I don't want the error grow with the number of
measurements or with the gain, that is related to the measruement device. If I
choose
+This is not what I want. I don't want the error grow with the number of
measurements or with the length of the reference direction, which is related to
the kind of the measurement. If I choose
\begin{equation}
-w_a = \frac{\norm a_2}{n_a \cdot \norm g_2} \quad and \quad w_m = \frac{\norm
m_2}{n_m \cdot \norm h_2}
+w_a = \frac{1}{n_a \cdot \norm g_2} \quad and \quad w_m = \frac{1}{n_m \cdot
\norm h_2}
\end{equation}
I get something like
\begin{equation}
@@ -155,7 +216,7 @@
That is an acceptable fact, since it helps to keep the matrix bound.
But because I want to do live-update of the attitude profile matrix I don't
know the real amount of measurements $n_a$ and $n_m$. But I know the
measurement frequencies $f_a$ and $f_m$, which are directly linked to them ($f
= \tfrac n T $). So my final decision for the measurement weight is
\begin{equation}
-w_a = \frac{\norm a_2}{f_a \cdot \norm g_2} \quad and \quad w_m = \frac{\norm
m_2}{f_m \cdot \norm h_2} \quad .
+w_a = \frac{1}{f_a \cdot \norm g_2} \quad and \quad w_m = \frac{1}{f_m \cdot
\norm h_2} \quad .
\end{equation}
The resulting error is then
\begin{equation}
Modified: paparazzi3/trunk/sw/airborne/fms/libeknav/doc/headfile.pdf
===================================================================
--- paparazzi3/trunk/sw/airborne/fms/libeknav/doc/headfile.pdf 2010-10-25
08:58:08 UTC (rev 6226)
+++ paparazzi3/trunk/sw/airborne/fms/libeknav/doc/headfile.pdf 2010-10-25
11:32:12 UTC (rev 6227)
@@ -134,36 +134,24 @@
/ProcSet [ /PDF /Text ]
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+/Length 2371
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