From 98b98d14d7e2326e841f155ecf790f9659034d27 Mon Sep 17 00:00:00 2001 From: Frank Dellaert Date: Thu, 4 Mar 2010 06:34:45 +0000 Subject: [PATCH] Substantial updates --- doc/LieGroups.lyx | 1631 ++++++++++++++++++++++++++++++++++++++++++++- doc/LieGroups.pdf | Bin 98840 -> 124611 bytes doc/math.lyx | 347 +++++++++- doc/math.pdf | Bin 114013 -> 115893 bytes 4 files changed, 1925 insertions(+), 53 deletions(-) diff --git a/doc/LieGroups.lyx b/doc/LieGroups.lyx index 1076188bf..55254d52f 100644 --- a/doc/LieGroups.lyx +++ b/doc/LieGroups.lyx @@ -58,10 +58,734 @@ Frank Dellaert \end_layout \begin_layout Standard -\begin_inset CommandInset include -LatexCommand include -filename "macros.lyx" +\begin_inset Note Comment +status open +\begin_layout Plain Layout +Derivatives +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\deriv}[2]{\frac{\partial#1}{\partial#2}} +{\frac{\partial#1}{\partial#2}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\at}[2]{#1\biggr\rvert_{#2}} +{#1\biggr\rvert_{#2}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Jac}[3]{ \at{\deriv{#1}{#2}} {#3} } +{\at{\deriv{#1}{#2}}{#3}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +Lie Groups +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\xhat}{\hat{x}} +{\hat{x}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\yhat}{\hat{y}} +{\hat{y}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Ad}[1]{Ad_{#1}} +{Ad_{#1}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\define}{\stackrel{\Delta}{=}} +{\stackrel{\Delta}{=}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\gg}{\mathfrak{g}} +{\mathfrak{g}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Rn}{\mathbb{R}^{n}} +{\mathbb{R}^{n}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SO(2), 1 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}} +{\mathfrak{\mathbb{R}^{2}}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\SOtwo}{SO(2)} +{SO(2)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\sotwo}{\mathfrak{so(2)}} +{\mathfrak{so(2)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\that}{\hat{\theta}} +{\hat{\theta}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\skew}[1]{[#1]_{+}} +{[#1]_{+}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SE(2), 3 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\SEtwo}{SE(2)} +{SE(2)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\setwo}{\mathfrak{se(2)}} +{\mathfrak{se(2)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Skew}[1]{[#1]_{\times}} +{[#1]_{\times}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SO(3), 3 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\Rthree}{\mathfrak{\mathbb{R}^{3}}} +{\mathfrak{\mathbb{R}^{3}}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\SOthree}{SO(3)} +{SO(3)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\sothree}{\mathfrak{so(3)}} +{\mathfrak{so(3)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\what}{\hat{\omega}} +{\hat{\omega}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SE(3),6 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\Rsix}{\mathfrak{\mathbb{R}^{6}}} +{\mathfrak{\mathbb{R}^{6}}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\SEthree}{SE(3)} +{SE(3)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\sethree}{\mathfrak{se(3)}} +{\mathfrak{se(3)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\xihat}{\hat{\xi}} +{\hat{\xi}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +Aff(2),6 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\Afftwo}{Aff(2)} +{Aff(2)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\afftwo}{\mathfrak{aff(2)}} +{\mathfrak{aff(2)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\aa}{a} +{a} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\ahat}{\hat{a}} +{\hat{a}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SL(3),8 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\SLthree}{SL(3)} +{SL(3)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\slthree}{\mathfrak{sl(3)}} +{\mathfrak{sl(3)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\hh}{h} +{h} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\hhat}{\hat{h}} +{\hat{h}} +\end_inset + + +\end_layout + +\begin_layout Section +Motivation: Rigid Motions in the Plane +\end_layout + +\begin_layout Standard +\begin_inset Float figure +placement h +wide false +sideways false +status collapsed + +\begin_layout Plain Layout +IMAGINE A FIGURE HERE +\begin_inset Caption + +\begin_layout Plain Layout +(a) A robot translating. + (b) A robot rotating. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +We will start with a small example of a robot moving in a plane, parameterized + by a +\emph on +2D pose +\emph default + +\begin_inset Formula $(x,\, y,\,\theta)$ +\end_inset + +. + When we give it a small forward velocity +\begin_inset Formula $v_{x}$ +\end_inset + +, we know that the location changes as +\begin_inset Formula \[ +\dot{x}=v_{x}\] + +\end_inset + +The solution to this trivial differential equation is, with +\begin_inset Formula $x_{0}$ +\end_inset + + the initial +\begin_inset Formula $x$ +\end_inset + +-position f the robot, +\begin_inset Formula \[ +x=x_{0}+v_{x}t\] + +\end_inset + +A similar story holds for translation in the +\begin_inset Formula $y$ +\end_inset + + direction, and in fact for translations in general: +\begin_inset Formula \[ +(x,\, y,\,\theta)=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0})\] + +\end_inset + +Similarly for rotation we have +\begin_inset Formula \[ +(x,\, y,\,\theta)=(x_{0},\, y_{0},\,\theta_{0}+\omega t)\] + +\end_inset + +where +\begin_inset Formula $\omega$ +\end_inset + + is angular velocity, measured in +\begin_inset Formula $rad/s$ +\end_inset + + in counterclockwise direction. +\end_layout + +\begin_layout Standard +\begin_inset Float figure +placement h +wide false +sideways false +status collapsed + +\begin_layout Plain Layout +IMAGINE A FIGURE HERE +\begin_inset Caption + +\begin_layout Plain Layout +Robot moving along a circular trajectory. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +However, if we combine translation and rotation, the story breaks down! + We cannot write +\begin_inset Formula \[ +(x,\, y,\,\theta)=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t)\] + +\end_inset + +The reason is that, if we move the robot a tiny bit according to the velocity + vector +\begin_inset Formula $(v_{x},\, v_{y},\,\omega)$ +\end_inset + +, we do have to first order +\begin_inset Formula \[ +(x_{t+\delta},\, y_{t+\delta},\,\theta_{t+\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)\] + +\end_inset + +but now the robot has rotated, and for the next incremental change, the + velocity vector would have to be rotated before it can be applied. + In fact, the robot will move on a +\emph on +circular +\emph default + trajectory. + +\end_layout + +\begin_layout Standard +The reason is that +\emph on +translation and rotation do not commute +\emph default +: if we rotate and then move we will end up in a different place than if + we moved first, then rotated. + In fact, someone once said (I forget who, kudos for who can track down + the exact quote): +\end_layout + +\begin_layout Quote +If rotation and translation commuted, we could do all rotations before leaving + home. +\end_layout + +\begin_layout Standard +To make progress, we have to be more precise about how the robot behaves. + Specifically, let us define composition of two poses +\begin_inset Formula $T_{1}$ +\end_inset + + and +\begin_inset Formula $T_{2}$ +\end_inset + + as +\begin_inset Formula \[ +T_{1}T_{2}=(x_{1},\, y_{1},\,\theta_{1})(x_{2},\, y_{2},\,\theta_{2})=(x_{1}+\cos\theta_{1}x_{2}-\sin\theta y_{2},\, y_{1}+\sin\theta_{1}x_{2}+\cos\theta_{1}y_{2},\,\theta_{1}+\theta_{2})\] + +\end_inset + +This is a bit clumsy, so we resort to a trick: embed the 2D poses in the + +\begin_inset Formula $3\times3$ +\end_inset + + matrices, so we can define composition as matrix multiplication: +\begin_inset Formula \[ +T_{1}T_{2}=\left[\begin{array}{cc} +R_{1} & t_{1}\\ +0 & 1\end{array}\right]\left[\begin{array}{cc} +R_{2} & t_{2}\\ +0 & 1\end{array}\right]=\left[\begin{array}{cc} +R_{1}R_{2} & R_{1}t_{2}+t_{1}\\ +0 & 1\end{array}\right]\] + +\end_inset + +where the matrices +\begin_inset Formula $R$ +\end_inset + + are 2D rotation matrices defined as +\begin_inset Formula \[ +R=\left[\begin{array}{cc} +\cos\theta & -\sin\theta\\ +\sin\theta & \cos\theta\end{array}\right]\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Float figure +placement h +wide false +sideways false +status open + +\begin_layout Plain Layout +IMAGINE A FIGURE HERE +\begin_inset Caption + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:n-step-program" + +\end_inset + +Approximating a circular trajectory with +\begin_inset Formula $n$ +\end_inset + + steps. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Now, a +\begin_inset Quotes eld +\end_inset + +tiny +\begin_inset Quotes erd +\end_inset + + motion of the robot can be written as +\begin_inset Formula \[ +T(\delta)=\left[\begin{array}{ccc} +\cos\omega\delta & -\sin\omega\delta & v_{x}\delta\\ +\sin\omega\delta & \cos\omega\delta & v_{y}\delta\\ +0 & 0 & 1\end{array}\right]\approx\left[\begin{array}{ccc} +1 & -\omega\delta & v_{x}\delta\\ +\omega\delta & 1 & v_{y}\delta\\ +0 & 0 & 1\end{array}\right]=I+\delta\left[\begin{array}{ccc} +0 & -\omega & v_{x}\\ +\omega & 0 & v_{y}\\ +0 & 0 & 1\end{array}\right]\] + +\end_inset + +Let us define the +\emph on +2D twist +\emph default + vector +\begin_inset Formula $\xi=(v,\omega)$ +\end_inset + +, and the matrix above as +\begin_inset Formula \[ +\xihat\define\left[\begin{array}{ccc} +0 & -\omega & v_{x}\\ +\omega & 0 & v_{y}\\ +0 & 0 & 1\end{array}\right]\] + +\end_inset + +If we wanted +\begin_inset Formula $t$ +\end_inset + + to be large, we could split up +\begin_inset Formula $t$ +\end_inset + + into smaller timesteps, say +\begin_inset Formula $n$ +\end_inset + + of them, and compose them as follows: +\begin_inset Formula \[ +T(t)\approx\left(I+\frac{t}{n}\xihat\right)\ldots\mbox{n times}\ldots\left(I+\frac{t}{n}\xihat\right)=\left(I+\frac{t}{n}\xihat\right)^{n}\] + +\end_inset + +The result is shown in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:n-step-program" + +\end_inset + +. +\end_layout + +\begin_layout Standard +Of course, the perfect solution would be obtained if we take +\begin_inset Formula $n$ +\end_inset + + to infinity: +\begin_inset Formula \[ +T(t)=\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}\] + +\end_inset + +For real numbers, this series is familiar and is actually a way to compute + the exponential function: +\begin_inset Formula \[ +e^{x}=\lim_{n\rightarrow\infty}\left(I+\frac{x}{n}\right)^{n}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\] + +\end_inset + +The series can be similarly defined for square matrices, +\begin_inset Formula \[ +e^{A}=\lim_{n\rightarrow\infty}\left(I+\frac{A}{n}\right)^{n}=\sum_{k=0}^{\infty}\frac{A^{k}}{k!}\] + +\end_inset + +Our final result is that we can write the motion of a robot along a circular + trajectory, resulting from the 2D twist +\begin_inset Formula $\xi=(v,\omega)$ +\end_inset + + +\begin_inset Formula $ $ +\end_inset + + as the matrix exponential of +\begin_inset Formula $\xihat$ +\end_inset + +: +\begin_inset Formula \[ +T(t)=e^{t\xihat}\define\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}=\sum_{k=0}^{\infty}\frac{\left(t\xihat\right)^{k}}{k!}\] + +\end_inset + +We call this mapping from 2D twists matrices +\begin_inset Formula $\xihat$ +\end_inset + + to 2D rigid transformations the +\emph on +exponential map. +\end_layout + +\begin_layout Standard +The above has all elemtns of Lie group theory. + We call the space of 2D rigid transformations, along with the composition + operation, the +\emph on +special Euclidean group +\emph default + +\begin_inset Formula $\SEtwo$ +\end_inset + +. + It is called a Lie group because it is both a manifold, and its group operation + is smooth when operating on this manifold. + The space of 2D twists, together with a special binary operation to be + defined below, is called the Lie algebra +\begin_inset Formula $\setwo$ +\end_inset + + associated with +\begin_inset Formula $\SEtwo$ +\end_inset + +. + Below we generalize these concepts and then introduce the most commonly + used Lie groups and their Lie algebras. +\end_layout + +\begin_layout Standard +\begin_inset Newpage pagebreak \end_inset @@ -101,13 +825,27 @@ A Lie group \end_inset -and a corresponding inverse +which is typically a many-to-one mapping. + The corresponding inverse can be define locally around the origin and hence + is a +\begin_inset Quotes eld +\end_inset + +logarithm +\begin_inset Quotes erd +\end_inset + +: \begin_inset Formula \[ \log:G\rightarrow\gg\] \end_inset -that maps elements in G to an element in +that maps elements in a neighborhood of +\begin_inset Formula $id$ +\end_inset + + in G to an element in \begin_inset Formula $\gg$ \end_inset @@ -222,15 +960,17 @@ For \begin_inset Formula $\mathbb{R}^{n}$ \end_inset -, and we can define the mapping -\begin_inset Formula \[ -\hat{}:\mathbb{R}^{n}\rightarrow\gg\] +, and we can define the wedge operator +\begin_inset CommandInset citation +LatexCommand cite +after "page 41" +key "Murray94book" \end_inset - +, \begin_inset Formula \[ -\hat{}:x\rightarrow\xhat\] +\hat{}:x\in\mathbb{R}^{n}\rightarrow\xhat\in\gg\] \end_inset @@ -239,11 +979,7 @@ which maps \end_inset -vectors -\begin_inset Formula $x\in$ -\end_inset - - -\begin_inset Formula $\Rn$ +\begin_inset Formula $x\in\mathbb{R}^{n}$ \end_inset to elements of @@ -286,7 +1022,7 @@ where the \begin_inset Formula $G$ \end_inset - and will be very intuitive. + and will generally have an intuitive interpretation. \end_layout \begin_layout Subsection @@ -348,7 +1084,7 @@ http://en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_group \begin_inset Formula \[ -\Ad T{\xhat}\define Te^{\xhat}T^{-1}\] +\Ad T{\xhat}\define T\xhat T^{-1}\] \end_inset @@ -362,11 +1098,11 @@ Te^{\xhat}T^{-1}=e^{T\xhat T^{-1}}\] \end_inset where both -\begin_inset Formula $T$ +\begin_inset Formula $T\in G$ \end_inset and -\begin_inset Formula $\xhat$ +\begin_inset Formula $\xhat\in\gg$ \end_inset are @@ -527,6 +1263,51 @@ R=e^{\skew{\theta}}=\left[\begin{array}{cc} \end_inset +\end_layout + +\begin_layout Subsection +Adjoint +\end_layout + +\begin_layout Standard +The adjoint map for +\begin_inset Formula $\sotwo$ +\end_inset + + is trivially equal to the identity, as is the case for +\emph on +all +\emph default + commutative groups: +\begin_inset Formula \begin{eqnarray*} +\Ad R\what & = & \left[\begin{array}{cc} +\cos\theta & -\sin\theta\\ +\sin\theta & \cos\theta\end{array}\right]\left[\begin{array}{cc} +0 & -\omega\\ +\omega & 0\end{array}\right]\left[\begin{array}{cc} +\cos\theta & -\sin\theta\\ +\sin\theta & \cos\theta\end{array}\right]^{T}\\ + & = & \omega\left[\begin{array}{cc} +-\sin\theta & -\cos\theta\\ +\cos\theta & -\sin\theta\end{array}\right]\left[\begin{array}{cc} +\cos\theta & \sin\theta\\ +-\sin\theta & \cos\theta\end{array}\right]=\left[\begin{array}{cc} +0 & -\omega\\ +\omega & 0\end{array}\right]\end{eqnarray*} + +\end_inset + +which we can write in terms of +\begin_inset Formula $\omega$ +\end_inset + + as +\begin_inset Formula \[ +\Ad R\omega=\omega\] + +\end_inset + + \end_layout \begin_layout Subsection @@ -633,7 +1414,76 @@ where \end_inset is a translation vector. - Its Lie algebra + +\begin_inset Formula $\SEtwo$ +\end_inset + + is the +\emph on +semi-direct product +\emph default + of +\begin_inset Formula $\Rtwo$ +\end_inset + + by +\begin_inset Formula $SO(2)$ +\end_inset + +, written as +\begin_inset Formula $\SEtwo=\Rtwo\rtimes\SOtwo$ +\end_inset + +. + In particular, any element +\begin_inset Formula $T$ +\end_inset + + of +\begin_inset Formula $\SEtwo$ +\end_inset + + can be written as +\begin_inset Formula \[ +T=\left[\begin{array}{cc} +0 & t\\ +0 & 1\end{array}\right]\left[\begin{array}{cc} +R & 0\\ +0 & k\end{array}\right]\] + +\end_inset + +and they compose as +\begin_inset Formula \[ +T_{1}T_{2}=\left[\begin{array}{cc} +R_{1} & t_{1}\\ +0 & 1\end{array}\right]\left[\begin{array}{cc} +R_{2} & t_{2}\\ +0 & 1\end{array}\right]=\left[\begin{array}{cc} +R_{1}R_{2} & R_{1}t_{2}+t_{1}\\ +0 & 1\end{array}\right]\] + +\end_inset + +Hence, an alternative way of writing down elements of +\begin_inset Formula $\SEtwo$ +\end_inset + + is as the ordered pair +\begin_inset Formula $(R,\, t)$ +\end_inset + +, with composition defined a +\begin_inset Formula \[ +(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1})\] + +\end_inset + + +\end_layout + +\begin_layout Standard +The corresponding Lie algebra \begin_inset Formula $\setwo$ \end_inset @@ -678,7 +1528,7 @@ Note we think of robots as having a pose \bar no \noun off \color none -The Lie group generators are +The corresponding Lie group generators are \begin_inset Formula \[ G^{x}=\left[\begin{array}{ccc} 0 & 0 & 1\\ @@ -712,11 +1562,11 @@ Applying the exponential map to a twist : \begin_inset Formula \[ -T=\exp\xihat\] +T=e^{\xihat}=\left(e^{\skew{\omega}},(I-e^{\skew{\omega}})\frac{v^{\perp}}{\omega}\right)\] \end_inset -A closed form solution for the exponential map is in the works... + \end_layout \begin_layout Subsection @@ -872,8 +1722,52 @@ The Lie group \begin_inset Formula $3\times3$ \end_inset - skew-symmetric matrices. - The exponential map can be computed in closed form using Rodrigues' formula. + skew-symmetric matrices +\begin_inset Formula $\what$ +\end_inset + +. + The exponential map can be computed in closed form using Rodrigues' formula + +\begin_inset CommandInset citation +LatexCommand cite +after "page 28" +key "Murray94book" + +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula \[ +e^{\what}=I+\what\sin\theta+\what^{2}(1\text{−}\cos\theta)\] + +\end_inset + +where +\begin_inset Formula $\what^{2}=\omega\omega^{T}-I$ +\end_inset + +, with +\begin_inset Formula $\omega\omega^{T}$ +\end_inset + + the outer product of +\begin_inset Formula $\omega$ +\end_inset + +. + Hence, a slightly more efficient variant is +\end_layout + +\begin_layout Standard +\begin_inset Formula \[ +e^{\what}=\cos\theta I+\what sin\theta+\omega\omega^{T}(1\text{−}cos\theta)\] + +\end_inset + + \end_layout \begin_layout Standard @@ -1017,7 +1911,7 @@ For rotation matrices we can prove the following identity (see \begin_inset CommandInset ref LatexCommand vref -reference "remove" +reference "proof1" \end_inset @@ -1030,7 +1924,7 @@ R\Skew{\omega}R^{T}=\Skew{R\omega}\label{eq:property1}\end{equation} Hence, given property \begin_inset CommandInset ref LatexCommand eqref -reference "remove" +reference "proof1" \end_inset @@ -1195,6 +2089,20 @@ where \end_inset is a translation vector. + An alternative way of writing down elements of +\begin_inset Formula $\SEthree$ +\end_inset + + is as the ordered pair +\begin_inset Formula $(R,\, t)$ +\end_inset + +, with composition defined as +\begin_inset Formula \[ +(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1})\] + +\end_inset + Its Lie algebra \begin_inset Formula $\sethree$ \end_inset @@ -1299,6 +2207,28 @@ key "Murray94book" . \end_layout +\begin_layout Standard + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\noun off +\color none +\begin_inset Formula \[ +\exp\left(\left[\begin{array}{c} +\omega\\ +v\end{array}\right]t\right)=\left[\begin{array}{cc} +e^{\Skew{\omega}t} & (I-e^{\Skew{\omega}t})\left(\omega\times v\right)+\omega\omega^{T}vt\\ +0 & 1\end{array}\right]\] + +\end_inset + + +\end_layout + \begin_layout Subsection The Adjoint Map \end_layout @@ -1458,13 +2388,560 @@ v\end{array}\right]=H_{p}\xi\] \end_inset +\end_layout + +\begin_layout Section +2D Affine Transformations +\end_layout + +\begin_layout Standard +The Lie group +\begin_inset Formula $Aff(2)$ +\end_inset + + is a subgroup of the general linear group +\begin_inset Formula $GL(3)$ +\end_inset + + of +\begin_inset Formula $3\times3$ +\end_inset + + invertible matrices that maps the line infinity to itself, and hence preserves + paralellism. + The affine transformation matrices +\begin_inset Formula $A$ +\end_inset + + can be written as +\begin_inset CommandInset citation +LatexCommand cite +key "Mei08tro" + +\end_inset + + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\noun off +\color none + +\begin_inset Formula \[ +\left[\begin{array}{ccc} +m_{11} & m_{12} & t_{1}\\ +m_{21} & m_{22} & t_{2}\\ +0 & 0 & k\end{array}\right]\] + +\end_inset + +with +\begin_inset Formula $M\in GL(2)$ +\end_inset + +, +\begin_inset Formula $t\in\Rtwo$ +\end_inset + +, and +\begin_inset Formula $k$ +\end_inset + + a scalar chosen such that +\begin_inset Formula $det(A)=1$ +\end_inset + +. + +\family default +\series default +\shape default +\size default +\emph default +\bar default +\noun default +\color inherit +Note that just as +\begin_inset Formula $\SEtwo$ +\end_inset + + is a semi-direct product, so too is +\begin_inset Formula $Aff(2)=\Rtwo\rtimes GL(2)$ +\end_inset + +. + In particular, any affine transformation +\begin_inset Formula $A$ +\end_inset + + can be written as +\begin_inset Formula \[ +A=\left[\begin{array}{cc} +0 & t\\ +0 & 1\end{array}\right]\left[\begin{array}{cc} +M & 0\\ +0 & k\end{array}\right]\] + +\end_inset + +and they compose as +\begin_inset Formula \[ +A_{1}A_{2}=\left[\begin{array}{cc} +M_{1} & t_{1}\\ +0 & k_{1}\end{array}\right]\left[\begin{array}{cc} +M_{2} & t_{2}\\ +0 & k_{2}\end{array}\right]=\left[\begin{array}{cc} +M_{1}M_{2} & M_{2}t_{2}+k_{2}t_{1}\\ +0 & k_{1}k_{2}\end{array}\right]\] + +\end_inset + +From this it can be gleaned that the groups +\begin_inset Formula $\SOtwo$ +\end_inset + + and +\begin_inset Formula $\SEtwo$ +\end_inset + + are both subgroups, with +\begin_inset Formula $\SOtwo\subset\SEtwo\subset\Afftwo$ +\end_inset + +. + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\noun off +\color none +By choosing the generators carefully we maintain this subgroup hierarchy. + In particular, +\begin_inset Formula $\setwo$ +\end_inset + + +\begin_inset Formula \[ +G^{1}=\left[\begin{array}{ccc} +0 & 0 & 1\\ +0 & 0 & 0\\ +0 & 0 & 0\end{array}\right]\mbox{ }G^{2}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & 0 & 1\\ +0 & 0 & 0\end{array}\right]\mbox{ }G^{3}=\left[\begin{array}{ccc} +0 & -1 & 0\\ +1 & 0 & 0\\ +0 & 0 & 0\end{array}\right]\] + +\end_inset + +can be extended to the +\family default +\series default +\shape default +\size default +\emph default +\bar default +\noun default +\color inherit +Lie algebra +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\noun off +\color none + +\begin_inset Formula $\afftwo$ +\end_inset + + using the three additional generators +\begin_inset Formula \[ +G^{4}=\left[\begin{array}{ccc} +0 & 1 & 0\\ +1 & 0 & 0\\ +0 & 0 & 0\end{array}\right]\mbox{ }G^{5}=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & -1 & 0\\ +0 & 0 & 0\end{array}\right]\mbox{ }G^{6}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & -1 & 0\\ +0 & 0 & 1\end{array}\right]\] + +\end_inset + + +\family default +\series default +\shape default +\size default +\emph default +\bar default +\noun default +\color inherit +Hence, the Lie algebra +\begin_inset Formula $\afftwo$ +\end_inset + + is the vector space of +\begin_inset Formula $3\times3$ +\end_inset + + incremental affine transformations +\begin_inset Formula $\ahat$ +\end_inset + + parameterized by 6 parameters +\begin_inset Formula $\aa\in\mathbb{R}^{6}$ +\end_inset + +, with the mapping +\begin_inset Formula \[ +\aa\rightarrow\ahat\define\left[\begin{array}{ccc} +a_{5} & a_{4}-a_{3} & a_{1}\\ +a_{4}+a_{3} & -a_{5}-a_{6} & a_{2}\\ +0 & 0 & a_{6}\end{array}\right]\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Note that +\begin_inset Formula $G_{5}$ +\end_inset + + and +\begin_inset Formula $G_{6}$ +\end_inset + + change the relative scale of +\begin_inset Formula $x$ +\end_inset + + and +\begin_inset Formula $y$ +\end_inset + + but without changing the determinant: +\begin_inset Formula \[ +e^{xG_{5}}=\exp\left(\left[\begin{array}{ccc} +x & 0 & 0\\ +0 & -x & 0\\ +0 & 0 & 0\end{array}\right]\right)=\left[\begin{array}{ccc} +e^{x} & 0 & 0\\ +0 & 1/e^{x} & 0\\ +0 & 0 & 1\end{array}\right]\] + +\end_inset + + +\begin_inset Formula \[ +e^{xG_{6}}=\exp\left(\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & -x & 0\\ +0 & 0 & x\end{array}\right]\right)=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & 1/e^{x} & 0\\ +0 & 0 & e^{x}\end{array}\right]\] + +\end_inset + +It might be nicer to have the correspondence with scaling +\begin_inset Formula $x$ +\end_inset + + and +\begin_inset Formula $y$ +\end_inset + + more direct, by choosing +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\noun off +\color none + +\begin_inset Formula \[ +\mbox{ }G^{5}=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & 0 & 0\\ +0 & 0 & -1\end{array}\right]\mbox{ }G^{6}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & 1 & 0\\ +0 & 0 & -1\end{array}\right]\] + +\end_inset + +and hence +\family default +\series default +\shape default +\size default +\emph default +\bar default +\noun default +\color inherit + +\begin_inset Formula \[ +e^{xG_{5}}=\exp\left(\left[\begin{array}{ccc} +x & 0 & 0\\ +0 & 0 & 0\\ +0 & 0 & -x\end{array}\right]\right)=\left[\begin{array}{ccc} +e^{x} & 0 & 0\\ +0 & 1 & 0\\ +0 & 0 & 1/e^{x}\end{array}\right]\] + +\end_inset + + +\begin_inset Formula \[ +e^{xG_{6}}=\exp\left(\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & x & 0\\ +0 & 0 & -x\end{array}\right]\right)=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & e^{x} & 0\\ +0 & 0 & 1/e^{x}\end{array}\right]\] + +\end_inset + + +\end_layout + +\begin_layout Section +2D Homographies +\end_layout + +\begin_layout Standard +When viewed as operations on images, represented by 2D projective space + +\begin_inset Formula $\mathcal{P}^{3}$ +\end_inset + +, 3D rotations are a special case of 2D homographies. + These are now treated, loosely based on the exposition in +\begin_inset CommandInset citation +LatexCommand cite +key "Mei06iros,Mei08tro" + +\end_inset + +. +\end_layout + +\begin_layout Subsection +Basics +\end_layout + +\begin_layout Standard +The Lie group +\begin_inset Formula $\SLthree$ +\end_inset + + is a subgroup of the general linear group +\begin_inset Formula $GL(3)$ +\end_inset + + of +\begin_inset Formula $3\times3$ +\end_inset + + invertible matrices with determinant +\begin_inset Formula $1$ +\end_inset + +. + The homographies generalize transformations of the 2D projective space, + and +\begin_inset Formula $\Afftwo\subset\SLthree$ +\end_inset + +. + +\end_layout + +\begin_layout Standard + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\noun off +\color none +We can extend +\begin_inset Formula $\afftwo$ +\end_inset + + to the Lie algebra +\begin_inset Formula $\slthree$ +\end_inset + + by adding two generators +\begin_inset Formula \[ +G^{7}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & 0 & 0\\ +1 & 0 & 0\end{array}\right]\mbox{ }G^{8}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & 0 & 0\\ +0 & 1 & 0\end{array}\right]\] + +\end_inset + + +\family default +\series default +\shape default +\size default +\emph default +\bar default +\noun default +\color inherit +obtaining the vector space of +\begin_inset Formula $3\times3$ +\end_inset + + incremental homographies +\begin_inset Formula $\hhat$ +\end_inset + + parameterized by 8 parameters +\begin_inset Formula $\hh\in\mathbb{R}^{8}$ +\end_inset + +, with the mapping +\begin_inset Formula \[ +h\rightarrow\hhat\define\left[\begin{array}{ccc} +h_{5} & h_{4}-h_{3} & h_{1}\\ +h_{4}+h_{3} & -h_{5}-h_{6} & h_{2}\\ +h_{7} & h_{8} & h_{6}\end{array}\right]\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +Tensor Notation +\end_layout + +\begin_layout Itemize +A homography between 2D projective spaces +\begin_inset Formula $A$ +\end_inset + + and +\begin_inset Formula $B$ +\end_inset + + can be written in tensor notation +\begin_inset Formula $H_{A}^{B}$ +\end_inset + + +\end_layout + +\begin_layout Itemize +Applying a homography is then a tensor contraction +\begin_inset Formula $x^{B}=H_{A}^{B}x^{A}$ +\end_inset + +, mapping points in +\begin_inset Formula $A$ +\end_inset + + to points in +\begin_inset Formula $B$ +\end_inset + +. +\end_layout + +\begin_layout Plain Layout +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +The inverse of a homography can be found by contracting with two permutation + tensors: +\begin_inset Formula \[ +H_{B}^{A}=H_{A_{1}}^{B_{1}}H_{A_{2}}^{B_{2}}\epsilon_{B_{1}B_{2}B}\epsilon^{A_{1}A_{2}A}\] + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Note Note +status collapsed + +\begin_layout Subsection +The Adjoint Map +\end_layout + +\begin_layout Plain Layout +The adjoint can be done using tensor notation. + Denoting an incremental homography in space +\begin_inset Formula $A$ +\end_inset + + as +\begin_inset Formula $\hhat_{A_{1}}^{A_{2}}$ +\end_inset + +, we have, for example for +\begin_inset Formula $G_{1}$ +\end_inset + + +\begin_inset Formula \begin{eqnarray*} +\hhat_{B_{1}}^{B_{2}}=\Ad{H_{A}^{B}}{\hhat_{A_{1}}^{A_{2}}} & = & H_{A_{2}}^{B_{2}}\hhat_{A_{1}}^{A_{2}}H_{B_{1}}^{A_{1}}\\ + & = & H_{A_{2}}^{B_{2}}\left[\begin{array}{ccc} +0 & 0 & 1\\ +0 & 0 & 0\\ +0 & 0 & 0\end{array}\right]H_{A_{2}}^{B_{2}}H_{A_{3}}^{B_{3}}\epsilon_{B_{1}B_{2}B_{3}}\epsilon^{A_{1}A_{2}A_{3}}\\ + & = & H_{1}^{B_{2}}H_{A_{2}}^{B_{2}}H_{A_{3}}^{B_{3}}\epsilon_{B_{1}B_{2}B_{3}}\epsilon^{3A_{2}A_{3}}\end{eqnarray*} + +\end_inset + +This does not seem to help. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Newpage pagebreak +\end_inset + + \end_layout \begin_layout Section* Appendix: Proof of Property \begin_inset CommandInset ref LatexCommand ref -reference "remove" +reference "proof1" \end_inset @@ -1494,7 +2971,7 @@ a_{3}(\omega\times a_{1}) & a_{3}(\omega\times a_{2}) & a_{3}(\omega\times a_{3} 0 & -\omega a_{3} & \omega a_{2}\\ \omega a_{3} & 0 & -\omega a_{1}\\ -\omega a_{2} & \omega a_{1} & 0\end{array}\right]\nonumber \\ - & = & \Skew{R\omega}\label{remove}\end{eqnarray} + & = & \Skew{R\omega}\label{proof1}\end{eqnarray} \end_inset @@ -1526,7 +3003,103 @@ a(b\times c)=b(c\times a)=c(a\times b)\] \end_inset +Similarly, without proof +\begin_inset CommandInset citation +LatexCommand cite +after "Lemma 2.3" +key "Murray94book" +\end_inset + +: +\begin_inset Formula \[ +R(a\times b)=Ra\times Rb\] + +\end_inset + + +\end_layout + +\begin_layout Section* +Appendix: Alternative Generators for +\begin_inset Formula $\slthree$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset CommandInset citation +LatexCommand cite +key "Mei06iros" + +\end_inset + + uses the following generators for +\begin_inset Formula $\slthree$ +\end_inset + +: +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\noun off +\color none + +\begin_inset Formula \[ +G^{1}=\left[\begin{array}{ccc} +0 & 0 & 1\\ +0 & 0 & 0\\ +0 & 0 & 0\end{array}\right]\mbox{ }G^{2}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & 0 & 1\\ +0 & 0 & 0\end{array}\right]\mbox{ }G^{3}=\left[\begin{array}{ccc} +0 & 1 & 0\\ +0 & 0 & 0\\ +0 & 0 & 0\end{array}\right]\] + +\end_inset + + +\begin_inset Formula \[ +G^{4}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +1 & 0 & 0\\ +0 & 0 & 0\end{array}\right]\mbox{ }G^{5}=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & -1 & 0\\ +0 & 0 & 0\end{array}\right]\mbox{ }G^{6}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & -1 & 0\\ +0 & 0 & 1\end{array}\right]\] + +\end_inset + + +\begin_inset Formula \[ +G^{7}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & 0 & 0\\ +1 & 0 & 0\end{array}\right]\mbox{ }G^{8}=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & 0 & 0\\ +0 & 1 & 0\end{array}\right]\] + +\end_inset + + +\family default +\series default +\shape default +\size default +\emph default +\bar default +\noun default +\color inherit +We choose to use a different linear combination as the basis. \end_layout 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+Derivatives +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\deriv}[2]{\frac{\partial#1}{\partial#2}} +{\frac{\partial#1}{\partial#2}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\at}[2]{#1\biggr\rvert_{#2}} +{#1\biggr\rvert_{#2}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Jac}[3]{ \at{\deriv{#1}{#2}} {#3} } +{\at{\deriv{#1}{#2}}{#3}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +Lie Groups +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\xhat}{\hat{x}} +{\hat{x}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\yhat}{\hat{y}} +{\hat{y}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Ad}[1]{Ad_{#1}} +{Ad_{#1}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\define}{\stackrel{\Delta}{=}} +{\stackrel{\Delta}{=}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\gg}{\mathfrak{g}} +{\mathfrak{g}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Rn}{\mathbb{R}^{n}} +{\mathbb{R}^{n}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SO(2) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}} +{\mathfrak{\mathbb{R}^{2}}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\SOtwo}{SO(2)} +{SO(2)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\sotwo}{\mathfrak{so(2)}} +{\mathfrak{so(2)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\that}{\hat{\theta}} +{\hat{\theta}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\skew}[1]{[#1]_{+}} +{[#1]_{+}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SE(2) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\SEtwo}{SE(2)} +{SE(2)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\setwo}{\mathfrak{se(2)}} +{\mathfrak{se(2)}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SO(3) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\Rthree}{\mathfrak{\mathbb{R}^{3}}} +{\mathfrak{\mathbb{R}^{3}}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\SOthree}{SO(3)} +{SO(3)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\sothree}{\mathfrak{so(3)}} +{\mathfrak{so(3)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\what}{\hat{\omega}} +{\hat{\omega}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Skew}[1]{[#1]_{\times}} +{[#1]_{\times}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +SE(3) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\Rsix}{\mathfrak{\mathbb{R}^{6}}} +{\mathfrak{\mathbb{R}^{6}}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\SEthree}{SE(3)} +{SE(3)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\sethree}{\mathfrak{se(3)}} +{\mathfrak{se(3)}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\xihat}{\hat{\xi}} +{\hat{\xi}} \end_inset @@ -289,6 +528,72 @@ Hence, now we undo frame. \end_layout +\begin_layout Subsection* +Numerical Derivatives +\end_layout + +\begin_layout Standard +Let's examine +\begin_inset Formula \[ +f\left(g\right)e^{\yhat}=f\left(ge^{\xhat}\right)\] + +\end_inset + +and multiply with +\begin_inset Formula $f(g)^{-1}$ +\end_inset + + on both sides then take the log (which in our case returns +\begin_inset Formula $y$ +\end_inset + +, not +\begin_inset Formula $\yhat$ +\end_inset + +): +\begin_inset Formula \[ +y(x)=\log\left[f\left(g\right)^{-1}f\left(ge^{\xhat}\right)\right]\] + +\end_inset + +Let us look at +\begin_inset Formula $x=0$ +\end_inset + +, and perturb in direction +\begin_inset Formula $i$ +\end_inset + +, +\begin_inset Formula $e_{i}=[0,0,d,0,0]$ +\end_inset + +. + Then take derivative, +\begin_inset Formula \[ +\deriv{y(d)}d\define\lim_{d->0}\frac{y(d)-y(0)}{d}=\lim_{d->0}\frac{1}{d}\log\left[f\left(g\right)^{-1}f\left(ge^{\hat{e_{i}}}\right)\right]\] + +\end_inset + +which is the basis for a numerical derivative scheme. +\end_layout + +\begin_layout Standard +Let us also look at a chain rule. + If we know the behavior at the origin +\begin_inset Formula $I$ +\end_inset + +, we can extrapolate +\begin_inset Formula \[ +f(ge^{\xhat})=f(ge^{\xhat}g^{-1}g)=f(e^{\Ad g\xhat}g)\] + +\end_inset + + +\end_layout + \begin_layout Section Derivatives of Actions \begin_inset CommandInset label @@ -594,15 +899,7 @@ Derivatives of Mappings \end_layout \begin_layout Standard -The adjoint map for -\begin_inset Formula $\sotwo$ -\end_inset - - is trivially equal to the identity, as is the case for -\emph on -all -\emph default - commutative groups, and we have the derivative of +We have the derivative of \series bold inverse \series default @@ -1065,10 +1362,10 @@ Hence, as with inverse \series default , -\begin_inset Formula \begin{eqnarray*} -\frac{\partial T^{-1}}{\partial\xi} & = & -\left[\begin{array}{cc} +\begin_inset Formula \[ +\frac{\partial T^{-1}}{\partial\xi}=\Ad T=-\left[\begin{array}{cc} R & 0\\ -\Skew tR & R\end{array}\right]\end{eqnarray*} +\Skew tR & R\end{array}\right]\] \end_inset @@ -1077,16 +1374,18 @@ R & 0\\ compose \series default in its first argument, -\begin_inset Formula \begin{eqnarray*} -\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{1}} & = & \left[\begin{array}{cc} +\begin_inset Formula \[ +\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{1}}=\Ad{T_{2}^{-1}}=\left[\begin{array}{cc} R_{2}^{T} & 0\\ -\Skew{-R_{2}^{T}t}R_{2}^{T} & R_{2}^{T}\end{array}\right]\end{eqnarray*} +\Skew{-R_{2}^{T}t_{2}}R_{2}^{T} & R_{2}^{T}\end{array}\right]=\left[\begin{array}{cc} +R_{2}^{T} & 0\\ +R_{2}^{T}\Skew{-t_{2}} & R_{2}^{T}\end{array}\right]\] \end_inset compose in its second argument, -\begin_inset Formula \begin{eqnarray*} -\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{2}} & = & I_{6}\end{eqnarray*} +\begin_inset Formula \[ +\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{2}}=I_{6}\] \end_inset @@ -1095,16 +1394,16 @@ compose in its second argument, between \series default in its first argument, -\begin_inset Formula \begin{eqnarray*} -\frac{\partial\left(T_{1}^{^{-1}}T_{2}\right)}{\partial\xi_{1}} & = & -\left[\begin{array}{cc} +\begin_inset Formula \[ +\frac{\partial\left(T_{1}^{^{-1}}T_{2}\right)}{\partial\xi_{1}}=\Ad{T_{21}}=-\left[\begin{array}{cc} R & 0\\ -\Skew tR & R\end{array}\right]\end{eqnarray*} +\Skew tR & R\end{array}\right]\] \end_inset with 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