From 84c25b234694248640756454a3000a4e8afacdd6 Mon Sep 17 00:00:00 2001 From: Frank Dellaert Date: Thu, 11 Mar 2010 21:39:44 +0000 Subject: [PATCH] inverse action, much nicer --- doc/math.lyx | 63 +++++++++++++++++++++++++++++++++++++++------------ doc/math.pdf | Bin 115893 -> 119284 bytes 2 files changed, 48 insertions(+), 15 deletions(-) diff --git a/doc/math.lyx b/doc/math.lyx index be415269d..fd620bcf2 100644 --- a/doc/math.lyx +++ b/doc/math.lyx @@ -54,7 +54,7 @@ Geometry Derivatives and Other Hairy Math Frank Dellaert \end_layout -\begin_layout Plain Layout +\begin_layout Standard \begin_inset Box Frameless position "t" hor_pos "c" @@ -326,7 +326,7 @@ Derivatives of Lie Group Mappings \end_layout \begin_layout Subsection -New +Homomorphisms \end_layout \begin_layout Standard @@ -410,6 +410,10 @@ e^{-\xhat}=\left(e^{-\xhat}\right)^{-1}\] \end_layout \begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout Let us define two mappings \begin_inset Formula \[ \Phi_{1}(A)=AB\mbox{ and }\Phi_{2}(B)=AB\] @@ -423,10 +427,15 @@ Then \end_inset +\end_layout + +\end_inset + + \end_layout \begin_layout Subsection -Old +Derivatives \end_layout \begin_layout Standard @@ -903,28 +912,22 @@ we would now like to know what an incremental action \begin_inset Formula \begin{eqnarray*} q(x) & = & \left(Te^{\xhat}\right)^{-1}p\\ & = & e^{-\xhat}T^{-1}p\\ - & = & T^{-1}Te^{-\xhat}T^{-1}p\\ - & = & -T^{-1}\exp\left(T\xhat T^{-1}\right)p\end{eqnarray*} + & = & e^{-\xhat}q\\ + & \approx & q-\xhat q\end{eqnarray*} \end_inset Hence \begin_inset Formula \begin{equation} -\deriv{q(x)}x=-T^{-1}\deriv{\left(T\xhat T^{-1}\mbox{ }p\right)}x\label{eq:inverseAction}\end{equation} +\deriv{q(x)}x=\deriv{\left(q-\xhat q\right)}x=-\deriv{\left(\xhat q\right)}x=-H_{q}\label{eq:inverseAction}\end{equation} \end_inset -The derivative in -\begin_inset Formula $p$ +where +\begin_inset Formula $H_{q}$ \end_inset - is again easy for matrix Lie groups: -\begin_inset Formula \[ -\deriv{\left(T^{-1}p\right)}p=T^{-1}\] - -\end_inset - - + will be as above. \end_layout \begin_layout Section @@ -1679,6 +1682,36 @@ in homogenous coordinates. \end_inset +The derivative of the inverse action +\begin_inset Formula $T^{-1}p$ +\end_inset + + is given by formula +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:inverseAction" + +\end_inset + +: +\end_layout + +\begin_layout Standard + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\noun off +\color none +\begin_inset Formula \[ +\deriv{\hat{q}(\xi)}{\xi}=-H_{q}=\left[\begin{array}{cc} +\Skew q & -I_{3}\end{array}\right]\] + +\end_inset + \end_layout diff --git a/doc/math.pdf b/doc/math.pdf index a36f8adc705ffd6cd9d3b89a75d983f0b7325bf1..98e3fd75807bc2c32f8f6ef3a5d91938a0ad596d 100644 GIT binary patch delta 95168 zcmbTdV~{7^+OJtwmu=g&ZQHil#edmdwr$&0UAApoU3Q(`JN7>3nVEM!%zVkXV?}1f zipa>du65nNY>t3@D1?k724&)6O9aqU0j9L195%#JdoDEUIIzM?8X2ISr50$#f(pVN 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