From 0f78d861ad24157f6039f349f551cca54fac03fc Mon Sep 17 00:00:00 2001 From: Frank Dellaert Date: Wed, 10 Mar 2010 00:12:05 +0000 Subject: [PATCH] version I gave to Jean Gallier --- doc/LieGroups.lyx | 21 +++++---------------- doc/LieGroups.pdf | Bin 124611 -> 128396 bytes 2 files changed, 5 insertions(+), 16 deletions(-) diff --git a/doc/LieGroups.lyx b/doc/LieGroups.lyx index e39b854b2..1df1c4a60 100644 --- a/doc/LieGroups.lyx +++ b/doc/LieGroups.lyx @@ -2042,15 +2042,11 @@ q(\omega)=Re^{\Skew{\omega}}p\] hence the derivative is: \begin_inset Formula \[ -\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)=R\deriv{}{\omega}\left(\Skew{\omega}p\right)=RH_{p}\] +\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)=R\deriv{}{\omega}\left(\Skew{\omega}p\right)=R\Skew{-p}\] \end_inset -To calculate -\begin_inset Formula $H_{p}$ -\end_inset - - we make use of +To show the last equality note that \begin_inset Formula \[ \Skew{\omega}p=\omega\times p=-p\times\omega=\Skew{-p}\omega\] @@ -2323,16 +2319,9 @@ We would now like to know what an incremental rotation parameterized by \end_inset -hence the derivative (following the exposition in Section -\begin_inset CommandInset ref -LatexCommand ref -reference "sec:Derivatives-of-Actions" - -\end_inset - -): +hence the derivative is \begin_inset Formula \[ -\deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right)=TH_{p}\] +\deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right)\] \end_inset @@ -2387,7 +2376,7 @@ By only taking the top three rows, we can write this as a velocity in \omega\times p+v=-p\times\omega+v=\left[\begin{array}{cc} -\Skew p & I_{3}\end{array}\right]\left[\begin{array}{c} \omega\\ -v\end{array}\right]=H_{p}\xi\] +v\end{array}\right]\] \end_inset diff --git a/doc/LieGroups.pdf b/doc/LieGroups.pdf index d4fff0299905085db1fbd663d01c07acf31d1be3..005b8c8357883301854f2efa51c474f4a7d06b6c 100644 GIT binary patch delta 108845 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