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[paparazzi-commits] [6167] added small documentation for the initialisat
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From: |
Martin Dieblich |
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Subject: |
[paparazzi-commits] [6167] added small documentation for the initialisation of the kalman-filter |
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Date: |
Mon, 18 Oct 2010 19:00:01 +0000 |
Revision: 6167
http://svn.sv.gnu.org/viewvc/?view=rev&root=paparazzi&revision=6167
Author: mdieblich
Date: 2010-10-18 19:00:00 +0000 (Mon, 18 Oct 2010)
Log Message:
-----------
added small documentation for the initialisation of the kalman-filter
Modified Paths:
--------------
paparazzi3/trunk/sw/airborne/fms/libeknav/estimate_attitude.c
paparazzi3/trunk/sw/airborne/fms/libeknav/libeknav_from_log.cpp
paparazzi3/trunk/sw/airborne/fms/libeknav/raw_log_to_ascii.c
Added Paths:
-----------
paparazzi3/trunk/sw/airborne/fms/libeknav/doc/
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
Added: paparazzi3/trunk/sw/airborne/fms/libeknav/doc/content.tex
===================================================================
--- paparazzi3/trunk/sw/airborne/fms/libeknav/doc/content.tex
(rev 0)
+++ paparazzi3/trunk/sw/airborne/fms/libeknav/doc/content.tex 2010-10-18
19:00:00 UTC (rev 6167)
@@ -0,0 +1,167 @@
+\section{ENU to NED transformations}
+I had the problem very often that I have to transform form ENU no NED. The
simple conversion: ``Flip x and y and negate z'' doesn't work for quaternions
or if you want to use matrix algebra.
+
+\subsection{Matrix}
+Flipping x and y and negating z is easy to express as a matrix:
+\begin{equation}
+R_{ENU2NED} = \begin{pmatrix}
+0 & 1 & 0 \\
+1 & 0 & 0 \\
+0 & 0 & -1
+\end{pmatrix}
+\end{equation}
+This works in both directions, since $ R_{ENU2NED} = \transp R_{ENU2NED} $.
+
+\subsection{Quaternion}
+It's easy to compute a quaternion out of the above rotation matrix.
+\begin{equation}
+\quat{ENU2NED} = \frac{1}{\sqrt 2} \begin{pmatrix}
+0 \\ 1 \\ 1 \\ 0
+\end{pmatrix} = \frac{1}{\sqrt 2} (i + j)
+\end{equation}
+This makes sense, since the real value = 0 represents a rotation about
$180^{\circ}$ and the three values for the axis $ \vect v =
\transp{\begin{pmatrix}1&1&0\end{pmatrix}}$ represent the axis of rotation.
+
+\subsubsection*{Transforming a quaternion between ENU/NED}
+If you want to cahnge a quaternion from NED to ENU or vice versa. It's not
totally simple like for vectors. \\
+If your quaternion consist of the values:
+\begin{equation}
+\quat{ECEF2NED} = \begin{pmatrix}a \\ b \\ c\\d\end{pmatrix}
+\end{equation}
+Then a transformation to ENU (NED) is made as following:
+\begin{equation}
+\quat{ECEF2NED} = \frac{1}{\sqrt 2} \begin{pmatrix}-b-c \\ a+d \\
a-d\\-b+c\end{pmatrix}
+\end{equation}
+
+Pay attention doing it twice! The multiplication of an NED to ENU quaternion
with itself leads to
+\begin{equation}
+\quat{ECEF2NED} \quatprod \quat{ECEF2NED} \quatprod \begin{pmatrix}a \\ b \\
c\\d\end{pmatrix}= \begin{pmatrix}-a \\ -b \\ -c\\-d\end{pmatrix} .
+\end{equation}
+This is logically the same rotation, but mathemaically a different quaternion.
So don't be confused if all values are negative :-)
+
+\section{Initialisation}
+
+\subsection{What about the standard deviation?}
+\begin{figure}[h!]\begin{center}
+ \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm,
+ semithick]
+ \tikzstyle{every state}=[draw=black,text=white]
+
+ \node (A) {Noise};
+ \node (B) [below of=A] {Measurement};
+ \node (C) [below of=B] {attitude profile matrix};
+ \node (D) [below of=C] {``K'' - matrix};
+ \node (E) [below of=D] {Eigenvector};
+ \node (F) [below of=E] {Euler angle};
+
+ \path (A) edge node {Gaussian noise} (B)
+ (B) edge node {weight of a single measurement} (C)
+ (C) edge node {(see the section below)} (D)
+ (D) edge node {computational error} (E)
+ (E) edge node {computational error} (F);
+
+ \end{tikzpicture}
+ \caption{Propagation of uncertainty}
+ \label{Propagation of uncertainty}
+\end{center}\end{figure}
+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).
+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}}
+\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.
+
+The resulting error is
+\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}
+
+The error for the ``K''-matrix is easy to get by inserting $ \mat B + \mat
{\sigma_B} $ into
+\begin{equation}
+\mat K = \begin{bmatrix}
+trace(\mat B) & \transp{\vect Z} \\
+\vect Z & \mat B + \transp{\mat B} - trace(\mat B) \mat I
+\end{bmatrix}
+\end{equation}
+
+
+
+
+
+
+
+
+
+
+
+\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
+\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
+\end{equation}
+and assume that $\mat B$ has an error $\mat B +\mat{\sigma_B} $, you will get
+\begin{equation}
+\mat B +\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} \frac{\vect a_k +
\vect{\sigma_a}}{\norm{a_k}_2} \cdot \transp{\vect{g}} + w_m \sum_{k=1}^{n_m}
\frac{\vect m_k + \vect{\sigma_m}}{\norm{m_k}_2} \cdot \transp{\vect{h}}
+\end{equation}
+\begin{equation}
+\mat B +\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} \frac{\vect a_k}{\norm{a_k}}_2
\cdot \transp{\vect{g}} + \frac{\vect{\sigma_a}}{\norm{a_k}}_2 \cdot
\transp{\vect{g}} + w_m \sum_{k=1}^{n_m} \frac{\vect m_k}{\norm{m_k}}_2 \cdot
\transp{\vect{h}} + \frac{\vect{\sigma_m}}{\norm{m_k}}_2 \cdot \transp{\vect{h}}
+\end{equation}
+\begin{equation}
+\mat B +\mat{\sigma_B} = \underbrace{w_a \sum_{k=1}^{n_a} \frac{\vect
a_k}{\norm{a_k}}_2\cdot \transp{\vect{g}} + w_m \sum_{k=1}^{n_m} \frac{\vect
m_k}{\norm{m_k}}_2 \cdot \transp{\vect{h}}}_{\mat B} + w_a \sum_{k=1}^{n_a}
\frac{\vect{\sigma_a}}{\norm{a_k}}_2 \cdot \transp{\vect{g}} + w_m
\sum_{k=1}^{n_m} \frac{\vect{\sigma_m}}{\norm{m_k}}_2 \cdot \transp{\vect{h}}
+\end{equation}
+\begin{equation}
+\mat{\sigma_B} = w_a \sum_{k=1}^{n_a} \frac{\vect{\sigma_a}}{\norm{a_k}}_2
\cdot \transp{\vect{g}} + w_m \sum_{k=1}^{n_m}
\frac{\vect{\sigma_m}}{\norm{m_k}}_2 \cdot \transp{\vect{h}}
+\end{equation}
+$\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:
+\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}}
+\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}
+\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
+\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 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
+\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}
+\end{equation}
+I get something like
+\begin{equation}
+\norm{\mat{\sigma_B}}_{F} \le \norm{\sigma_a}_2 + \norm{\sigma_m}_2 \quad ,
+\end{equation}
+which looks much better. For the Frobenius norm of the attitude profile matrix
the choosen weight leads to
+\begin{equation}
+\norm{\mat B}_{F} \le \frac{1}{n_a} \sum_{k=1}^{n_a} \norm{a_k}_2 +
\frac{1}{n_m} \sum_{k=1}^{n_m} \norm{m_k}_2 \quad .
+\end{equation}
+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 .
+\end{equation}
+The resulting error is then
+\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}
+or
+\begin{equation}
+\norm{\mat{\sigma_B}}_{F} \le \frac{n_a}{f_a} \norm{\sigma_a}_2 +
\frac{n_m}{f_m} \norm{\sigma_m}_2 \quad ,
+\end{equation}
\ No newline at end of file
Added: paparazzi3/trunk/sw/airborne/fms/libeknav/doc/headfile.pdf
===================================================================
--- paparazzi3/trunk/sw/airborne/fms/libeknav/doc/headfile.pdf
(rev 0)
+++ paparazzi3/trunk/sw/airborne/fms/libeknav/doc/headfile.pdf 2010-10-18
19:00:00 UTC (rev 6167)
@@ -0,0 +1,2259 @@
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- [paparazzi-commits] [6167] added small documentation for the initialisation of the kalman-filter,
Martin Dieblich <=