From 6e1879a60d1ddf34cc0b147652e13cec054de2eb Mon Sep 17 00:00:00 2001 From: dellaert Date: Sun, 7 Feb 2016 22:25:51 -0800 Subject: [PATCH] Added Sim3 --- doc/LieGroups.lyx | 261 ++++++++++++++++++++++++++++++++++++++++++---- doc/LieGroups.pdf | Bin 319323 -> 267262 bytes 2 files changed, 241 insertions(+), 20 deletions(-) diff --git a/doc/LieGroups.lyx b/doc/LieGroups.lyx index 0d194b31d..bc3fb9780 100644 --- a/doc/LieGroups.lyx +++ b/doc/LieGroups.lyx @@ -1,5 +1,5 @@ -#LyX 2.0 created this file. For more info see http://www.lyx.org/ -\lyxformat 413 +#LyX 2.1 created this file. For more info see http://www.lyx.org/ +\lyxformat 474 \begin_document \begin_header \textclass article @@ -15,13 +15,13 @@ theorems-std \font_roman times \font_sans default \font_typewriter default +\font_math auto \font_default_family rmdefault \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 - \graphics default \default_output_format default \output_sync 0 @@ -32,15 +32,24 @@ theorems-std \use_hyperref false \papersize default \use_geometry true -\use_amsmath 1 -\use_esint 0 -\use_mhchem 1 -\use_mathdots 1 +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 0 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 \cite_engine basic +\cite_engine_type default +\biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false +\justification true \use_refstyle 0 \index Index \shortcut idx @@ -96,7 +105,7 @@ We will start with a small example of a robot moving in a plane, parameterized 2D pose \emph default -\begin_inset Formula $(x,\, y,\,\theta)$ +\begin_inset Formula $(x,\,y,\,\theta)$ \end_inset . @@ -135,7 +144,7 @@ A similar story holds for translation in the direction, and in fact for translations in general: \begin_inset Formula \[ -(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}) +(x_{t},\,y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\,y_{0}+v_{y}t,\,\theta_{0}) \] \end_inset @@ -143,7 +152,7 @@ A similar story holds for translation in the Similarly for rotation we have \begin_inset Formula \[ -(x_{t},\, y_{t},\,\theta_{t})=(x_{0},\, y_{0},\,\theta_{0}+\omega t) +(x_{t},\,y_{t},\,\theta_{t})=(x_{0},\,y_{0},\,\theta_{0}+\omega t) \] \end_inset @@ -175,7 +184,7 @@ status collapsed \end_inset -\begin_inset Caption +\begin_inset Caption Standard \begin_layout Plain Layout Robot moving along a circular trajectory. @@ -196,20 +205,20 @@ However, if we combine translation and rotation, the story breaks down! We cannot write \begin_inset Formula \[ -(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t) +(x_{t},\,y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\,y_{0}+v_{y}t,\,\theta_{0}+\omega t) \] \end_inset The reason is that, if we move the robot a tiny bit according to the velocity vector -\begin_inset Formula $(v_{x},\, v_{y},\,\omega)$ +\begin_inset Formula $(v_{x},\,v_{y},\,\omega)$ \end_inset , we have (to first order) \begin_inset Formula \[ -(x_{\delta},\, y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta) +(x_{\delta},\,y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\,y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta) \] \end_inset @@ -255,7 +264,7 @@ status open \end_inset -\begin_inset Caption +\begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label @@ -290,7 +299,7 @@ To make progress, we have to be more precise about how the robot behaves. as \begin_inset Formula \[ -T_{1}T_{2}=(x_{1},\, y_{1},\,\theta_{1})(x_{2},\, y_{2},\,\theta_{2})=(x_{1}+\cos\theta_{1}x_{2}-\sin\theta y_{2},\, y_{1}+\sin\theta_{1}x_{2}+\cos\theta_{1}y_{2},\,\theta_{1}+\theta_{2}) +T_{1}T_{2}=(x_{1},\,y_{1},\,\theta_{1})(x_{2},\,y_{2},\,\theta_{2})=(x_{1}+\cos\theta_{1}x_{2}-\sin\theta y_{2},\,y_{1}+\sin\theta_{1}x_{2}+\cos\theta_{1}y_{2},\,\theta_{1}+\theta_{2}) \] \end_inset @@ -1600,13 +1609,13 @@ Hence, an alternative way of writing down elements of \end_inset is as the ordered pair -\begin_inset Formula $(R,\, t)$ +\begin_inset Formula $(R,\,t)$ \end_inset , with composition defined a \begin_inset Formula \[ -(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1}) +(R_{1},\,t_{1})(R_{2},\,t_{2})=(R_{1}R_{2},\,R{}_{1}t_{2}+t_{1}) \] \end_inset @@ -2569,13 +2578,13 @@ where \end_inset is as the ordered pair -\begin_inset Formula $(R,\, t)$ +\begin_inset Formula $(R,\,t)$ \end_inset , with composition defined as \begin_inset Formula \[ -(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1}) +(R_{1},\,t_{1})(R_{2},\,t_{2})=(R_{1}R_{2},\,R{}_{1}t_{2}+t_{1}) \] \end_inset @@ -2944,6 +2953,218 @@ p\\ \end_inset +\end_layout + +\begin_layout Section +3D Similarity Transformations +\end_layout + +\begin_layout Standard +The group of 3D similarity transformations +\begin_inset Formula $Sim(3)$ +\end_inset + + is the set of +\begin_inset Formula $4\times4$ +\end_inset + + invertible matrices of the form +\begin_inset Formula +\[ +T\define\left[\begin{array}{cc} +R & t\\ +0 & s^{-1} +\end{array}\right] +\] + +\end_inset + +where +\begin_inset Formula $s$ +\end_inset + + is a scalar. + There are several different conventions in use for the Lie algebra generators, + but we use +\begin_inset Formula +\[ +G^{1}=\left(\begin{array}{cccc} +0 & 0 & 0 & 0\\ +0 & 0 & -1 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 0 +\end{array}\right)\mbox{}G^{2}=\left(\begin{array}{cccc} +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & 0 +\end{array}\right)\mbox{ }G^{3}=\left(\begin{array}{cccc} +0 & -1 & 0 & 0\\ +1 & 0 & 0 & 0\\ +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 0 +\end{array}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +G^{4}=\left(\begin{array}{cccc} +0 & 0 & 0 & 1\\ +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 0 +\end{array}\right)\mbox{}G^{5}=\left(\begin{array}{cccc} +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 1\\ +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 0 +\end{array}\right)\mbox{ }G^{6}=\left(\begin{array}{cccc} +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 1\\ +0 & 0 & 0 & 0 +\end{array}\right)\mbox{ }G^{7}=\left(\begin{array}{cccc} +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 0\\ +0 & 0 & 0 & 0\\ +0 & 0 & 0 & -1 +\end{array}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +Actions +\end_layout + +\begin_layout Standard +The action of +\begin_inset Formula $\SEthree$ +\end_inset + + on 3D points is done by embedding the points in +\begin_inset Formula $\mathbb{R}^{4}$ +\end_inset + + by using homogeneous coordinates +\begin_inset Formula +\[ +\hat{q}=\left[\begin{array}{c} +q\\ +s^{-1} +\end{array}\right]=\left[\begin{array}{c} +Rp+t\\ +s^{-1} +\end{array}\right]=\left[\begin{array}{cc} +R & t\\ +0 & s^{-1} +\end{array}\right]\left[\begin{array}{c} +p\\ +1 +\end{array}\right]=T\hat{p} +\] + +\end_inset + +The derivative +\begin_inset Formula $D_{1}f(\xi)$ +\end_inset + + in an incremental change +\begin_inset Formula $\xi$ +\end_inset + + to +\begin_inset Formula $T$ +\end_inset + + is given by +\begin_inset Formula $TH(p)$ +\end_inset + + where +\begin_inset Formula +\[ +H(p)=G_{jk}^{i}p^{j}=\left(\begin{array}{ccccccc} +0 & z & -y & 1 & 0 & 0 & 0\\ +-z & 0 & x & 0 & 1 & 0 & 0\\ +y & -x & 0 & 0 & 0 & 1 & 0\\ +0 & 0 & 0 & 0 & 0 & 0 & -1 +\end{array}\right) +\] + +\end_inset + +In other words +\begin_inset Formula +\[ +D_{1}f(\xi)=\left[\begin{array}{cc} +R & t\\ +0 & s^{-1} +\end{array}\right]\left[\begin{array}{ccc} +-\left[p\right]_{x} & I_{3} & 0\\ +0 & 0 & -1 +\end{array}\right]=\left[\begin{array}{ccc} +-R\left[p\right]_{x} & R & -t\\ +0 & 0 & -s^{-1} +\end{array}\right] +\] + +\end_inset + +This is the derivative for the action on homogeneous coordinates. + Switching back to non-homogeneous coordinates is done by +\begin_inset Formula +\[ +\left[\begin{array}{c} +q\\ +a +\end{array}\right]\rightarrow q/a +\] + +\end_inset + +with derivative +\begin_inset Formula +\[ +\left[\begin{array}{cc} +a^{-1}I_{3} & -qa^{-2}\end{array}\right] +\] + +\end_inset + +For +\begin_inset Formula $a=s^{-1}$ +\end_inset + + we obtain +\begin_inset Formula +\[ +D_{1}f(\xi)=\left[\begin{array}{cc} +sI_{3} & -qs^{2}\end{array}\right]\left[\begin{array}{ccc} +-R\left[p\right]_{x} & R & -t\\ +0 & 0 & -s^{-1} +\end{array}\right]=\left[\begin{array}{ccc} +-sR\left[p\right]_{x} & sR & -st+qs\end{array}\right]=\left[\begin{array}{ccc} +-sR\left[p\right]_{x} & sR & sRp\end{array}\right] +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Newpage 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