Added Sim3
dellaert
parent
a056086ea4
commit
6e1879a60d
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
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\lyxformat 413
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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@ -15,13 +15,13 @@ theorems-std
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\font_roman times
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\font_sans default
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\output_sync 0
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@ -32,15 +32,24 @@ theorems-std
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\use_hyperref false
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\papersize default
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\use_geometry true
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\use_amsmath 1
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\use_esint 0
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\use_mhchem 1
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\use_mathdots 1
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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\use_package esint 0
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\use_package stmaryrd 1
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\use_package undertilde 1
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\cite_engine basic
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\cite_engine_type default
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\paperorientation portrait
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\suppress_date false
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\justification true
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\use_refstyle 0
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\index Index
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\shortcut idx
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@ -96,7 +105,7 @@ We will start with a small example of a robot moving in a plane, parameterized
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2D pose
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\emph default
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\begin_inset Formula $(x,\, y,\,\theta)$
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\begin_inset Formula $(x,\,y,\,\theta)$
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\end_inset
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.
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@ -135,7 +144,7 @@ A similar story holds for translation in the
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direction, and in fact for translations in general:
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\begin_inset Formula
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\[
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(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0})
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(x_{t},\,y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\,y_{0}+v_{y}t,\,\theta_{0})
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\]
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\end_inset
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@ -143,7 +152,7 @@ A similar story holds for translation in the
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Similarly for rotation we have
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\begin_inset Formula
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\[
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(x_{t},\, y_{t},\,\theta_{t})=(x_{0},\, y_{0},\,\theta_{0}+\omega t)
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(x_{t},\,y_{t},\,\theta_{t})=(x_{0},\,y_{0},\,\theta_{0}+\omega t)
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\]
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\end_inset
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@ -175,7 +184,7 @@ status collapsed
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\end_inset
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\begin_inset Caption
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Robot moving along a circular trajectory.
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@ -196,20 +205,20 @@ However, if we combine translation and rotation, the story breaks down!
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We cannot write
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\begin_inset Formula
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\[
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(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t)
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(x_{t},\,y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\,y_{0}+v_{y}t,\,\theta_{0}+\omega t)
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\]
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\end_inset
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The reason is that, if we move the robot a tiny bit according to the velocity
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vector
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\begin_inset Formula $(v_{x},\, v_{y},\,\omega)$
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\begin_inset Formula $(v_{x},\,v_{y},\,\omega)$
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\end_inset
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, we have (to first order)
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\begin_inset Formula
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\[
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(x_{\delta},\, y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)
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(x_{\delta},\,y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\,y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)
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\]
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\end_inset
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@ -255,7 +264,7 @@ status open
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\end_inset
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\begin_inset Caption
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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\begin_inset CommandInset label
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@ -290,7 +299,7 @@ To make progress, we have to be more precise about how the robot behaves.
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as
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\begin_inset Formula
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\[
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T_{1}T_{2}=(x_{1},\, y_{1},\,\theta_{1})(x_{2},\, y_{2},\,\theta_{2})=(x_{1}+\cos\theta_{1}x_{2}-\sin\theta y_{2},\, y_{1}+\sin\theta_{1}x_{2}+\cos\theta_{1}y_{2},\,\theta_{1}+\theta_{2})
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T_{1}T_{2}=(x_{1},\,y_{1},\,\theta_{1})(x_{2},\,y_{2},\,\theta_{2})=(x_{1}+\cos\theta_{1}x_{2}-\sin\theta y_{2},\,y_{1}+\sin\theta_{1}x_{2}+\cos\theta_{1}y_{2},\,\theta_{1}+\theta_{2})
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\]
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\end_inset
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@ -1600,13 +1609,13 @@ Hence, an alternative way of writing down elements of
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\end_inset
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is as the ordered pair
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\begin_inset Formula $(R,\, t)$
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\begin_inset Formula $(R,\,t)$
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\end_inset
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, with composition defined a
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\begin_inset Formula
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\[
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(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1})
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(R_{1},\,t_{1})(R_{2},\,t_{2})=(R_{1}R_{2},\,R{}_{1}t_{2}+t_{1})
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\]
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\end_inset
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@ -2569,13 +2578,13 @@ where
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\end_inset
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is as the ordered pair
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\begin_inset Formula $(R,\, t)$
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\begin_inset Formula $(R,\,t)$
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\end_inset
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, with composition defined as
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\begin_inset Formula
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\[
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(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1})
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(R_{1},\,t_{1})(R_{2},\,t_{2})=(R_{1}R_{2},\,R{}_{1}t_{2}+t_{1})
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\]
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\end_inset
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@ -2944,6 +2953,218 @@ p\\
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\end_inset
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\end_layout
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\begin_layout Section
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3D Similarity Transformations
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\end_layout
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\begin_layout Standard
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The group of 3D similarity transformations
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\begin_inset Formula $Sim(3)$
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\end_inset
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is the set of
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\begin_inset Formula $4\times4$
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\end_inset
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invertible matrices of the form
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\begin_inset Formula
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\[
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T\define\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]
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\]
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\end_inset
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where
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\begin_inset Formula $s$
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\end_inset
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is a scalar.
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There are several different conventions in use for the Lie algebra generators,
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but we use
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\begin_inset Formula
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\[
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G^{1}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & -1 & 0\\
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0 & 1 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{}G^{2}=\left(\begin{array}{cccc}
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0 & 0 & 1 & 0\\
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0 & 0 & 0 & 0\\
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-1 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{3}=\left(\begin{array}{cccc}
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0 & -1 & 0 & 0\\
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1 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)
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\]
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\end_inset
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\begin_inset Formula
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\[
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G^{4}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{}G^{5}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{6}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0
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\end{array}\right)\mbox{ }G^{7}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & -1
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\end{array}\right)
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Actions
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\end_layout
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\begin_layout Standard
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The action of
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\begin_inset Formula $\SEthree$
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\end_inset
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on 3D points is done by embedding the points in
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\begin_inset Formula $\mathbb{R}^{4}$
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\end_inset
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by using homogeneous coordinates
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\begin_inset Formula
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\[
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\hat{q}=\left[\begin{array}{c}
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q\\
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s^{-1}
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\end{array}\right]=\left[\begin{array}{c}
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Rp+t\\
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s^{-1}
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\end{array}\right]=\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]\left[\begin{array}{c}
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p\\
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1
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\end{array}\right]=T\hat{p}
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\]
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\end_inset
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The derivative
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\begin_inset Formula $D_{1}f(\xi)$
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\end_inset
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in an incremental change
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\begin_inset Formula $\xi$
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\end_inset
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to
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\begin_inset Formula $T$
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\end_inset
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is given by
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\begin_inset Formula $TH(p)$
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\end_inset
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where
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\begin_inset Formula
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\[
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H(p)=G_{jk}^{i}p^{j}=\left(\begin{array}{ccccccc}
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0 & z & -y & 1 & 0 & 0 & 0\\
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-z & 0 & x & 0 & 1 & 0 & 0\\
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y & -x & 0 & 0 & 0 & 1 & 0\\
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0 & 0 & 0 & 0 & 0 & 0 & -1
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\end{array}\right)
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\]
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\end_inset
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In other words
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\begin_inset Formula
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\[
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D_{1}f(\xi)=\left[\begin{array}{cc}
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R & t\\
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0 & s^{-1}
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\end{array}\right]\left[\begin{array}{ccc}
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-\left[p\right]_{x} & I_{3} & 0\\
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0 & 0 & -1
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\end{array}\right]=\left[\begin{array}{ccc}
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-R\left[p\right]_{x} & R & -t\\
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0 & 0 & -s^{-1}
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\end{array}\right]
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\]
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\end_inset
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This is the derivative for the action on homogeneous coordinates.
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Switching back to non-homogeneous coordinates is done by
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\begin_inset Formula
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\[
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\left[\begin{array}{c}
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q\\
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a
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\end{array}\right]\rightarrow q/a
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\]
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\end_inset
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with derivative
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\begin_inset Formula
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\[
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\left[\begin{array}{cc}
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a^{-1}I_{3} & -qa^{-2}\end{array}\right]
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\]
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\end_inset
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For
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\begin_inset Formula $a=s^{-1}$
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\end_inset
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we obtain
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\begin_inset Formula
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\[
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D_{1}f(\xi)=\left[\begin{array}{cc}
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sI_{3} & -qs^{2}\end{array}\right]\left[\begin{array}{ccc}
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-R\left[p\right]_{x} & R & -t\\
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0 & 0 & -s^{-1}
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\end{array}\right]=\left[\begin{array}{ccc}
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-sR\left[p\right]_{x} & sR & -st+qs\end{array}\right]=\left[\begin{array}{ccc}
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-sR\left[p\right]_{x} & sR & sRp\end{array}\right]
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Newpage pagebreak
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\end_inset
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\end_layout
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\begin_layout Section
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