#LyX 2.1 created this file. For more info see http://www.lyx.org/ \lyxformat 474 \begin_document \begin_header \textclass article \use_default_options false \begin_modules theorems-std \end_modules \maintain_unincluded_children false \language english \language_package default \inputencoding auto \fontencoding global \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 \bibtex_command default \index_command default \paperfontsize 12 \spacing single \use_hyperref false \papersize default \use_geometry true \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 \color #008000 \end_index \leftmargin 1in \topmargin 1in \rightmargin 1in \bottommargin 1in \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Title Lie Groups for Beginners \end_layout \begin_layout Author Frank Dellaert \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand include filename "macros.lyx" \end_inset \end_layout \begin_layout Section Motivation: Rigid Motions in the Plane \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 of the robot, \begin_inset Formula \[ x_{t}=x_{0}+v_{x}t \] \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 \[ (x_{t},\,y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\,y_{0}+v_{y}t,\,\theta_{0}) \] \end_inset Similarly for rotation we have \begin_inset Formula \[ (x_{t},\,y_{t},\,\theta_{t})=(x_{0},\,y_{0},\,\theta_{0}+\omega t) \] \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 \align center \begin_inset Graphics filename images/circular.pdf \end_inset \begin_inset Caption Standard \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 \[ (x_{t},\,y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\,y_{0}+v_{y}t,\,\theta_{0}+\omega t) \] \end_inset The reason is that, if we move the robot a tiny bit according to the velocity vector \begin_inset Formula $(v_{x},\,v_{y},\,\omega)$ \end_inset , we have (to first order) \begin_inset Formula \[ (x_{\delta},\,y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\,y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta) \] \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 \begin_inset Float figure placement h wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename images/n-steps.pdf \end_inset \begin_inset Caption Standard \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 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 space of \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 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 \[ T(\delta)=\left[\begin{array}{ccc} \cos\omega\delta & -\sin\omega\delta & v_{x}\delta\\ \sin\omega\delta & \cos\omega\delta & v_{y}\delta\\ 0 & 0 & 1 \end{array}\right]\approx\left[\begin{array}{ccc} 1 & -\omega\delta & v_{x}\delta\\ \omega\delta & 1 & v_{y}\delta\\ 0 & 0 & 1 \end{array}\right]=I+\delta\left[\begin{array}{ccc} 0 & -\omega & v_{x}\\ \omega & 0 & v_{y}\\ 0 & 0 & 0 \end{array}\right] \] \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 \[ \xihat\define\left[\begin{array}{ccc} 0 & -\omega & v_{x}\\ \omega & 0 & v_{y}\\ 0 & 0 & 0 \end{array}\right] \] \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 \[ e^{x}=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!} \] \end_inset The series can be similarly defined for square matrices, and the 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 \emph on matrix exponential \emph default of \begin_inset Formula $\xihat$ \end_inset : \begin_inset Formula \[ T(t)=e^{t\xihat}\define\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\xihat^{k} \] \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 elements 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 simultaneously a topological group and a manifold, which implies that the multiplication and the inversion operations are smooth. 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 . \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Section Basic Lie Group Concepts \end_layout \begin_layout Standard We now define the concepts illustrated above, introduce some notation, and see what we can say in general. After this we then introduce the most commonly used Lie groups and their Lie algebras. \end_layout \begin_layout Subsection A Manifold and a Group \end_layout \begin_layout Standard A \series bold Lie group \series default \begin_inset Formula $G$ \end_inset is both a group \emph on and \emph default a manifold that possesses a smooth group operation. Associated with it is a \series bold Lie Algebra \series default \begin_inset Formula $\gg$ \end_inset which, loosely speaking, can be identified with the tangent space at the identity and completely defines how the groups behaves around the identity. There is a mapping from \begin_inset Formula $\gg$ \end_inset back to \begin_inset Formula $G$ \end_inset , called the \series bold exponential map \series default \begin_inset Formula \[ \exp:\gg\rightarrow G \] \end_inset 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 a neighborhood of \begin_inset Formula $id$ \end_inset in G to an element in \begin_inset Formula $\gg$ \end_inset . \end_layout \begin_layout Standard An important family of Lie groups are the matrix Lie groups, whose elements are \begin_inset Formula $n\times n$ \end_inset invertible matrices. The set of all such matrices, together with the matrix multiplication, is called the general linear group \begin_inset Formula $GL(n)$ \end_inset of dimension \begin_inset Formula $n$ \end_inset , and any closed subgroup of it is a \series bold matrix Lie group \series default . Most if not all Lie groups we are interested in will be matrix Lie groups. \end_layout \begin_layout Subsection Lie Algebra \end_layout \begin_layout Standard The Lie Algebra \begin_inset Formula $\gg$ \end_inset is called an algebra because it is endowed with a binary operation, the \series bold Lie bracket \series default \begin_inset Formula $[X,Y]$ \end_inset , the properties of which are closely related to the group operation of \begin_inset Formula $G$ \end_inset . For example, for algebras associated with matrix Lie groups, the Lie bracket is given by \begin_inset Formula $[A,B]\define AB-BA$ \end_inset . \end_layout \begin_layout Standard The relationship of the Lie bracket to the group operation is as follows: for commutative Lie groups vector addition \begin_inset Formula $X+Y$ \end_inset in \begin_inset Formula $\gg$ \end_inset mimicks the group operation. For example, if we have \begin_inset Formula $Z=X+Y$ \end_inset in \begin_inset Formula $\gg$ \end_inset , when mapped backed to \begin_inset Formula $G$ \end_inset via the exponential map we obtain \begin_inset Formula \[ e^{Z}=e^{X+Y}=e^{X}e^{Y} \] \end_inset However, this does \emph on not \emph default hold for non-commutative Lie groups: \begin_inset Formula \[ Z=\log(e^{X}e^{Y})\neq X+Y \] \end_inset Instead, \begin_inset Formula $Z$ \end_inset can be calculated using the Baker-Campbell-Hausdorff (BCH) formula \begin_inset CommandInset citation LatexCommand cite key "Hall00book" \end_inset \begin_inset Note Note status collapsed \begin_layout Plain Layout http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula \end_layout \end_inset : \begin_inset Formula \[ Z=X+Y+[X,Y]/2+[X-Y,[X,Y]]/12-[Y,[X,[X,Y]]]/24+\ldots \] \end_inset For commutative groups the bracket is zero and we recover \begin_inset Formula $Z=X+Y$ \end_inset . For non-commutative groups we can use the BCH formula to approximate it. \end_layout \begin_layout Subsection Exponential Coordinates \end_layout \begin_layout Standard For \begin_inset Formula $n$ \end_inset -dimensional matrix Lie groups, as a vector space the Lie algebra \begin_inset Formula $\gg$ \end_inset is isomorphic to \begin_inset Formula $\mathbb{R}^{n}$ \end_inset , and we can define the hat operator \begin_inset CommandInset citation LatexCommand cite after "page 41" key "Murray94book" \end_inset , \begin_inset Formula \[ \hat{}:x\in\mathbb{R}^{n}\rightarrow\xhat\in\gg \] \end_inset which maps \begin_inset Formula $n$ \end_inset -vectors \begin_inset Formula $x\in\mathbb{R}^{n}$ \end_inset to elements of \begin_inset Formula $\gg$ \end_inset . In the case of matrix Lie groups, the elements \begin_inset Formula $\xhat$ \end_inset of \begin_inset Formula $\gg$ \end_inset are also \begin_inset Formula $n\times n$ \end_inset matrices, and the map is given by \begin_inset Formula \begin{equation} \xhat=\sum_{i=1}^{n}x_{i}G^{i}\label{eq:generators} \end{equation} \end_inset where the \begin_inset Formula $G^{i}$ \end_inset are \begin_inset Formula $n\times n$ \end_inset matrices known as Lie group generators. The meaning of the map \begin_inset Formula $x\rightarrow\xhat$ \end_inset will depend on the group \begin_inset Formula $G$ \end_inset and will generally have an intuitive interpretation. \end_layout \begin_layout Subsection Actions \end_layout \begin_layout Standard An important concept is that of a group element acting on an element of a manifold \begin_inset Formula $M$ \end_inset . For example, 2D rotations act on 2D points, 3D rotations act on 3D points, etc. In particular, a \series bold left action \series default of \begin_inset Formula $G$ \end_inset on \begin_inset Formula $M$ \end_inset is defined as a smooth map \begin_inset Formula $\Phi:G\times M\rightarrow M$ \end_inset such that \begin_inset CommandInset citation LatexCommand cite after "Appendix A" key "Murray94book" \end_inset : \end_layout \begin_layout Enumerate The identity element \begin_inset Formula $e$ \end_inset has no effect, i.e., \begin_inset Formula $\Phi(e,p)=p$ \end_inset \end_layout \begin_layout Enumerate Composing two actions can be combined into one action: \begin_inset Formula $\Phi(g,\Phi(h,p))=\Phi(gh,p)$ \end_inset \end_layout \begin_layout Standard The (usual) action of an \begin_inset Formula $n$ \end_inset -dimensional matrix group \begin_inset Formula $G$ \end_inset is matrix-vector multiplication on \begin_inset Formula $\mathbb{R}^{n}$ \end_inset , \begin_inset Formula \[ q=Ap \] \end_inset with \begin_inset Formula $p,q\in\mathbb{R}^{n}$ \end_inset and \begin_inset Formula $A\in G\subseteq GL(n)$ \end_inset . \end_layout \begin_layout Subsection The Adjoint Map and Adjoint Representation \end_layout \begin_layout Standard Suppose a point \begin_inset Formula $p$ \end_inset is specified as \begin_inset Formula $p'$ \end_inset in the frame \begin_inset Formula $T$ \end_inset , i.e., \begin_inset Formula $p'=Tp$ \end_inset , where \begin_inset Formula $T$ \end_inset transforms from the global coordinates \begin_inset Formula $p$ \end_inset to the local frame \begin_inset Formula $p'$ \end_inset . To apply an action \begin_inset Formula $A$ \end_inset we first need to undo \begin_inset Formula $T$ \end_inset , then apply \begin_inset Formula $A$ \end_inset , and then transform the result back to \begin_inset Formula $T$ \end_inset : \begin_inset Formula \[ q'=TAT^{-1}p' \] \end_inset The matrix \begin_inset Formula $TAT^{-1}$ \end_inset is said to be conjugate to \begin_inset Formula $A$ \end_inset , and this is a central element of group theory. \end_layout \begin_layout Standard In general, the \series bold adjoint map \series default \begin_inset Formula $\AAdd g$ \end_inset maps a group element \begin_inset Formula $a\in G$ \end_inset to its \series bold conjugate \series default \begin_inset Formula $gag^{-1}$ \end_inset by \begin_inset Formula $g$ \end_inset . This map captures conjugacy in the group \begin_inset Formula $G$ \end_inset , but there is an equivalent notion in the Lie algebra \begin_inset Formula $\mathfrak{\gg}$ \end_inset , \begin_inset Note Note status open \begin_layout Plain Layout http://en.wikipedia.org/wiki/Exponential_map \end_layout \end_inset \begin_inset Formula \[ \AAdd ge^{\xhat}=g\exp\left(\xhat\right)g^{-1}=\exp(\Ad g{\xhat}) \] \end_inset where \begin_inset Formula $\Ad g:\gg\rightarrow\mathfrak{\gg}$ \end_inset is a map parameterized by a group element \begin_inset Formula $g$ \end_inset , and is called the \emph on adjoint representation \emph default . The intuitive explanation is that a change \begin_inset Formula $\exp\left(\xhat\right)$ \end_inset defined around the origin, but applied at the group element \begin_inset Formula $g$ \end_inset , can be written in one step by taking the adjoint \begin_inset Formula $\Ad g{\xhat}$ \end_inset of \begin_inset Formula $\xhat$ \end_inset . \end_layout \begin_layout Standard In the special case of matrix Lie groups the adjoint can be written as \begin_inset Note Note status collapsed \begin_layout Plain Layout http://en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_group \end_layout \end_inset \begin_inset Formula \[ \Ad T{\xhat}\define T\xhat T^{-1} \] \end_inset and hence we have \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} Te^{\xhat}T^{-1}=e^{T\xhat T^{-1}}\label{eq:matrixAdjoint} \end{equation} \end_inset where both \begin_inset Formula $T\in G$ \end_inset and \begin_inset Formula $\xhat\in\gg$ \end_inset are \begin_inset Formula $n\times n$ \end_inset matrices for an \begin_inset Formula $n$ \end_inset -dimensional Lie group. \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Section 2D Rotations \end_layout \begin_layout Standard We first look at a very simple group, the 2D rotations. \end_layout \begin_layout Subsection Basics \end_layout \begin_layout Standard The Lie group \begin_inset Formula $\SOtwo$ \end_inset is a subgroup of the general linear group \begin_inset Formula $GL(2)$ \end_inset of \begin_inset Formula $2\times2$ \end_inset invertible matrices. Its Lie algebra \begin_inset Formula $\sotwo$ \end_inset is the vector space of \begin_inset Formula $2\times2$ \end_inset skew-symmetric matrices. Since \begin_inset Formula $\SOtwo$ \end_inset is a one-dimensional manifold, \begin_inset Formula $\sotwo$ \end_inset is isomorphic to \begin_inset Formula $\mathbb{R}$ \end_inset and we define \begin_inset Formula \[ \hat{}:\mathbb{R}\rightarrow\sotwo \] \end_inset \begin_inset Formula \[ \hat{}:\omega\rightarrow\what=\skew{\omega} \] \end_inset which maps the angle \begin_inset Formula $\omega$ \end_inset to the \begin_inset Formula $2\times2$ \end_inset skew-symmetric matrix \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula $\skew{\omega}$ \end_inset : \family default \series default \shape default \size default \emph default \bar default \noun default \color inherit \begin_inset Formula \[ \skew{\omega}=\left[\begin{array}{cc} 0 & -\omega\\ \omega & 0 \end{array}\right] \] \end_inset The exponential map can be computed in closed form as \begin_inset Formula \[ e^{\skew{\omega}}=\left[\begin{array}{cc} \cos\omega & -\sin\omega\\ \sin\omega & \cos\omega \end{array}\right] \] \end_inset \end_layout \begin_layout Subsection \begin_inset CommandInset label LatexCommand label name "sub:Diagonalized2D" \end_inset Diagonalized Form \end_layout \begin_layout Standard The matrix \begin_inset Formula $\skew 1$ \end_inset can be diagonalized (see \begin_inset CommandInset citation LatexCommand cite key "Hall00book" \end_inset ) with eigenvalues \begin_inset Formula $-i$ \end_inset and \begin_inset Formula $i$ \end_inset , and eigenvectors \begin_inset Formula $\left[\begin{array}{c} 1\\ i \end{array}\right]$ \end_inset and \begin_inset Formula $\left[\begin{array}{c} i\\ 1 \end{array}\right]$ \end_inset . Readers familiar with projective geometry will recognize these as the circular points when promoted to homogeneous coordinates. In particular: \begin_inset Formula \[ \skew{\omega}=\left[\begin{array}{cc} 0 & -\omega\\ \omega & 0 \end{array}\right]=\left[\begin{array}{cc} 1 & i\\ i & 1 \end{array}\right]\left[\begin{array}{cc} -i\omega & 0\\ 0 & i\omega \end{array}\right]\left[\begin{array}{cc} 1 & i\\ i & 1 \end{array}\right]^{-1} \] \end_inset and hence \begin_inset Formula \[ e^{\skew{\omega}}=\frac{1}{2}\left[\begin{array}{cc} 1 & i\\ i & 1 \end{array}\right]\left[\begin{array}{cc} e^{-i\omega} & 0\\ 0 & e^{i\omega} \end{array}\right]\left[\begin{array}{cc} 1 & -i\\ -i & 1 \end{array}\right]=\left[\begin{array}{cc} \cos\omega & -\sin\omega\\ \sin\omega & \cos\omega \end{array}\right] \] \end_inset where the latter can be shown using \begin_inset Formula $e^{i\omega}=\cos\omega+i\sin\omega$ \end_inset . \end_layout \begin_layout Subsection Adjoint \end_layout \begin_layout Standard The adjoint 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 i.e., \begin_inset Formula \[ \Ad R\what=\what \] \end_inset \end_layout \begin_layout Subsection Actions \end_layout \begin_layout Standard In the case of \begin_inset Formula $\SOtwo$ \end_inset the vector space is \begin_inset Formula $\Rtwo$ \end_inset , and the group action corresponds to rotating a point \begin_inset Formula \[ q=Rp \] \end_inset We would now like to know what an incremental rotation parameterized by \begin_inset Formula $\omega$ \end_inset would do: \begin_inset Formula \[ q(\text{\omega})=Re^{\skew{\omega}}p \] \end_inset For small angles \begin_inset Formula $\omega$ \end_inset we have \begin_inset Formula \[ e^{\skew{\omega}}\approx I+\skew{\omega}=I+\omega\skew 1 \] \end_inset where \begin_inset Formula $\skew 1$ \end_inset acts like a restricted \begin_inset Quotes eld \end_inset cross product \begin_inset Quotes erd \end_inset in the plane on points: \begin_inset Formula \begin{equation} \skew 1\left[\begin{array}{c} x\\ y \end{array}\right]=R_{\pi/2}\left[\begin{array}{c} x\\ y \end{array}\right]=\left[\begin{array}{c} -y\\ x \end{array}\right]\label{eq:RestrictedCross} \end{equation} \end_inset Hence the derivative of the action is given as \begin_inset Formula \[ \deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\skew{\omega}}p\right)=R\deriv{}{\omega}\left(\omega\skew 1p\right)=RH_{p} \] \end_inset where \begin_inset Formula $H_{p}$ \end_inset is a \begin_inset Formula $2\times1$ \end_inset matrix that depends on \begin_inset Formula $p$ \end_inset : \begin_inset Formula \[ H_{p}\define\skew 1p=\left[\begin{array}{c} -p_{y}\\ p_{x} \end{array}\right] \] \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Section 2D Rigid Transformations \end_layout \begin_layout Subsection Basics \end_layout \begin_layout Standard The Lie group \begin_inset Formula $\SEtwo$ \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 of the form \begin_inset Formula \[ T\define\left[\begin{array}{cc} R & t\\ 0 & 1 \end{array}\right] \] \end_inset where \begin_inset Formula $R\in\SOtwo$ \end_inset is a rotation matrix and \begin_inset Formula $t\in\Rtwo$ \end_inset is a translation vector. \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 \[ T=\left[\begin{array}{cc} 0 & t\\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} R & 0\\ 0 & 1 \end{array}\right] \] \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 is the vector space of \begin_inset Formula $3\times3$ \end_inset twists \begin_inset Formula $\xihat$ \end_inset parameterized by the \emph on twist coordinates \emph default \begin_inset Formula $\xi\in\Rthree$ \end_inset , with the mapping \begin_inset Formula \[ \xi\define\left[\begin{array}{c} v\\ \omega \end{array}\right]\rightarrow\xihat\define\left[\begin{array}{cc} \skew{\omega} & v\\ 0 & 0 \end{array}\right] \] \end_inset Note we think of robots as having a pose \begin_inset Formula $(x,y,\theta)$ \end_inset and hence I reserved the first two components for translation and the last for rotation. \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none The corresponding Lie group generators are \begin_inset Formula \[ G^{x}=\left[\begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]\mbox{ }G^{y}=\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{array}\right]\mbox{ }G^{\theta}=\left[\begin{array}{ccc} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right] \] \end_inset \family default \series default \shape default \size default \emph default \bar default \noun default \color inherit Applying the exponential map to a twist \begin_inset Formula $\xi$ \end_inset yields a screw motion yielding an element in \begin_inset Formula $\SEtwo$ \end_inset : \begin_inset Formula \[ T=e^{\xihat}=\left(e^{\skew{\omega}},(I-e^{\skew{\omega}})\frac{v^{\perp}}{\omega}\right) \] \end_inset \end_layout \begin_layout Subsection The Adjoint Map \end_layout \begin_layout Standard The adjoint is \begin_inset Formula \begin{eqnarray} \Ad T{\xihat} & = & T\xihat T^{-1}\nonumber \\ & = & =\left[\begin{array}{cc} R & t\\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} \skew{\omega} & v\\ 0 & 0 \end{array}\right]\left[\begin{array}{cc} R^{T} & -R^{T}t\\ 0 & 1 \end{array}\right]\nonumber \\ & = & \left[\begin{array}{cc} \skew{\omega} & -\skew{\omega}t+Rv\\ 0 & 0 \end{array}\right]\nonumber \\ & = & \left[\begin{array}{cc} \skew{\omega} & Rv-t^{\perp}\omega\\ 0 & 0 \end{array}\right]\label{eq:adjointSE2} \end{eqnarray} \end_inset From this we can express the Adjoint map in terms of plane twist coordinates: \begin_inset Formula \[ \left[\begin{array}{c} v'\\ \omega' \end{array}\right]=\left[\begin{array}{cc} R & -t^{\perp}\\ 0 & 1 \end{array}\right]\left[\begin{array}{c} v\\ \omega \end{array}\right] \] \end_inset \end_layout \begin_layout Subsection Actions \end_layout \begin_layout Standard The action of \begin_inset Formula $\SEtwo$ \end_inset on 2D points is done by embedding the points in \begin_inset Formula $\mathbb{R}^{3}$ \end_inset by using homogeneous coordinates \begin_inset Formula \[ \hat{q}=\left[\begin{array}{c} q\\ 1 \end{array}\right]=\left[\begin{array}{cc} R & t\\ 0 & 1 \end{array}\right]\left[\begin{array}{c} p\\ 1 \end{array}\right]=T\hat{p} \] \end_inset Analoguous to \begin_inset Formula $\SEthree$ \end_inset (see below), we can compute a velocity \begin_inset Formula $\xihat\hat{p}$ \end_inset in the local \begin_inset Formula $T$ \end_inset frame: \begin_inset Formula \[ \xihat\hat{p}=\left[\begin{array}{cc} \skew{\omega} & v\\ 0 & 0 \end{array}\right]\left[\begin{array}{c} p\\ 1 \end{array}\right]=\left[\begin{array}{c} \skew{\omega}p+v\\ 0 \end{array}\right] \] \end_inset By only taking the top two rows, we can write this as a velocity in \begin_inset Formula $\Rtwo$ \end_inset , as the product of a \begin_inset Formula $2\times3$ \end_inset matrix \begin_inset Formula $H_{p}$ \end_inset that acts upon the exponential coordinates \begin_inset Formula $\xi$ \end_inset directly: \begin_inset Formula \[ \skew{\omega}p+v=v+R_{\pi/2}p\omega=\left[\begin{array}{cc} I_{2} & R_{\pi/2}p\end{array}\right]\left[\begin{array}{c} v\\ \omega \end{array}\right]=H_{p}\xi \] \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Section 3D Rotations \end_layout \begin_layout Subsection Basics \end_layout \begin_layout Standard The Lie group \begin_inset Formula $\SOthree$ \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. Its Lie algebra \begin_inset Formula $\sothree$ \end_inset is the vector space of \begin_inset Formula $3\times3$ \end_inset skew-symmetric matrices \begin_inset Formula $\what$ \end_inset . Since \begin_inset Formula $\SOthree$ \end_inset is a three-dimensional manifold, \begin_inset Formula $\sothree$ \end_inset is isomorphic to \begin_inset Formula $\Rthree$ \end_inset and we define the map \begin_inset Formula \[ \hat{}:\Rthree\rightarrow\sothree \] \end_inset \begin_inset Formula \[ \hat{}:\omega\rightarrow\what=\Skew{\omega} \] \end_inset which maps 3-vectors \begin_inset Formula $\omega$ \end_inset to skew-symmetric matrices \begin_inset Formula $\Skew{\omega}$ \end_inset : \begin_inset Formula \[ \Skew{\omega}=\left[\begin{array}{ccc} 0 & -\omega_{z} & \omega_{y}\\ \omega_{z} & 0 & -\omega_{x}\\ -\omega_{y} & \omega_{x} & 0 \end{array}\right]=\omega_{x}G^{x}+\omega_{y}G^{y}+\omega_{z}G^{z} \] \end_inset Here the matrices \begin_inset Formula $G^{i}$ \end_inset are the generators for \begin_inset Formula $\SOthree$ \end_inset , \begin_inset Formula \[ G^{x}=\left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{array}\right)\mbox{}G^{y}=\left(\begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0\\ -1 & 0 & 0 \end{array}\right)\mbox{ }G^{z}=\left(\begin{array}{ccc} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right) \] \end_inset corresponding to a rotation around \begin_inset Formula $X$ \end_inset , \begin_inset Formula $Y$ \end_inset , and \begin_inset Formula $Z$ \end_inset , respectively. The Lie bracket \begin_inset Formula $[x,y]$ \end_inset in \begin_inset Formula $\sothree$ \end_inset corresponds to the cross product \begin_inset Formula $x\times y$ \end_inset in \begin_inset Formula $\Rthree$ \end_inset . \end_layout \begin_layout Standard Hence, for every \begin_inset Formula $3$ \end_inset -vector \begin_inset Formula $\omega$ \end_inset there is a corresponding rotation matrix \begin_inset Formula \[ R=e^{\Skew{\omega}} \] \end_inset which defines a canonical parameterization of \begin_inset Formula $\SOthree$ \end_inset , with \begin_inset Formula $\omega$ \end_inset known as the canonical or exponential coordinates. It is equivalent to the axis-angle representation for rotations, where the unit vector \begin_inset Formula $\omega/\theta$ \end_inset defines the rotation axis, and its magnitude the amount of rotation \begin_inset Formula $\theta$ \end_inset . \end_layout \begin_layout Standard The exponential map can be computed in closed form using \series bold Rodrigues' formula \series default \begin_inset CommandInset citation LatexCommand cite after "page 28" key "Murray94book" \end_inset : \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} e^{\what}=I+\frac{\sin\theta}{\theta}\what+\frac{1\text{−}\cos\theta}{\theta^{2}}\what^{2}\label{eq:Rodrigues} \end{equation} \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 \begin_inset Formula \begin{equation} e^{\what}=\left(\cos\theta\right)I+\frac{\sin\theta}{\theta}\what+\frac{1\text{−}cos\theta}{\theta^{2}}\omega\omega^{T}\label{eq:Rodrigues2} \end{equation} \end_inset \end_layout \begin_layout Subsection Diagonalized Form \end_layout \begin_layout Standard Because a 3D rotation \begin_inset Formula $R$ \end_inset leaves the axis \begin_inset Formula $\omega$ \end_inset unchanged, \begin_inset Formula $R$ \end_inset can be diagonalized as \begin_inset Formula \[ R=C\left(\begin{array}{ccc} e^{-i\theta} & 0 & 0\\ 0 & e^{i\theta} & 0\\ 0 & 0 & 1 \end{array}\right)C^{-1} \] \end_inset with \begin_inset Formula $C=\left(\begin{array}{ccc} c_{1} & c_{2} & \omega/\theta\end{array}\right)$ \end_inset , where \begin_inset Formula $c_{1}$ \end_inset and \begin_inset Formula $c_{2}$ \end_inset are the complex eigenvectors corresponding to the 2D rotation around \begin_inset Formula $\omega$ \end_inset . This also means that, by \begin_inset CommandInset ref LatexCommand eqref reference "eq:matrixAdjoint" \end_inset , \begin_inset Formula \[ \hat{\omega}=C\left(\begin{array}{ccc} -i\theta & 0 & 0\\ 0 & i\theta & 0\\ 0 & 0 & 0 \end{array}\right)C^{-1} \] \end_inset In this case, \begin_inset Formula $C$ \end_inset has complex columns, but we also have \begin_inset Formula \begin{equation} \hat{\omega}=B\left(\begin{array}{ccc} 0 & -\theta & 0\\ \theta & 0 & 0\\ 0 & 0 & 0 \end{array}\right)B^{T}\label{eq:OmegaDecomposed} \end{equation} \end_inset with \begin_inset Formula $B=\left(\begin{array}{ccc} b_{1} & b_{2} & \omega/\theta\end{array}\right)$ \end_inset , where \begin_inset Formula $b_{1}$ \end_inset and \begin_inset Formula $b_{2}$ \end_inset form a basis for the 2D plane through the origin and perpendicular to \begin_inset Formula $\omega$ \end_inset . Clearly, from Section \begin_inset CommandInset ref LatexCommand ref reference "sub:Diagonalized2D" \end_inset , we have \begin_inset Formula \[ c_{1}=B\left(\begin{array}{c} 1\\ i\\ 0 \end{array}\right)\mbox{\,\,\,\ and\,\,\,\,\,}c_{2}=B\left(\begin{array}{c} i\\ 1\\ 0 \end{array}\right) \] \end_inset and when we exponentiate \begin_inset CommandInset ref LatexCommand eqref reference "eq:OmegaDecomposed" \end_inset we expose the 2D rotation around the axis \begin_inset Formula $\omega/\theta$ \end_inset with magnitude \begin_inset Formula $\theta$ \end_inset : \begin_inset Formula \[ R=B\left(\begin{array}{ccc} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{array}\right)B^{T} \] \end_inset The latter form for \begin_inset Formula $R$ \end_inset can be used to prove Rodrigues' formula. Expanding the above, we get \begin_inset Formula \[ R=\left(\cos\theta\right)\left(b_{1}b_{1}^{T}+b_{2}b_{2}^{T}\right)+\left(\sin\theta\right)\left(b_{2}b_{1}^{T}-b_{1}b_{2}^{T}\right)+\omega\omega^{T}/\theta^{2} \] \end_inset \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \uuline off \uwave off \noun off \color none \begin_inset Note Note status collapsed \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \uuline off \uwave off \noun off \color none \begin_inset Formula \begin{eqnarray*} R & = & \left(\begin{array}{ccc} b_{1} & b_{2} & \omega/\theta\end{array}\right)\left(\begin{array}{ccc} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} b_{1}^{T}\\ b_{2}^{T}\\ \omega^{T}/\theta \end{array}\right)\\ & = & \left(\begin{array}{ccc} b_{1} & b_{2} & \omega/\theta\end{array}\right)\left(\begin{array}{c} b_{1}^{T}\cos\theta-b_{2}^{T}\sin\theta\\ b_{1}^{T}\sin\theta+b_{2}^{T}\cos\theta\\ \omega^{T}/\theta \end{array}\right)\\ & = & b_{1}b_{1}^{T}\cos\theta-b_{1}b_{2}^{T}\sin\theta+b_{2}b_{1}^{T}\sin\theta+b_{2}b_{2}^{T}\cos\theta+\omega\omega^{T}/\theta^{2} \end{eqnarray*} \end_inset \end_layout \end_inset Because \begin_inset Formula $B$ \end_inset is a rotation matrix, we have \begin_inset Formula $BB^{T}=b_{1}b_{1}^{T}+b_{2}b_{2}^{T}+\omega\omega^{T}/\theta^{2}=I$ \end_inset , and using \begin_inset CommandInset ref LatexCommand eqref reference "eq:OmegaDecomposed" \end_inset it is easy to show that \begin_inset Formula $b_{2}b_{1}^{T}-b_{1}b_{2}^{T}=\hat{\omega}/\theta$ \end_inset , hence \family default \series default \shape default \size default \emph default \bar default \strikeout default \uuline default \uwave default \noun default \color inherit \begin_inset Formula \[ R=\left(\cos\theta\right)(I-\omega\omega^{T}/\theta^{2})+\left(\sin\theta\right)\left(\hat{\omega}/\theta\right)+\omega\omega^{T}/\theta^{2} \] \end_inset which is equivalent to \begin_inset CommandInset ref LatexCommand eqref reference "eq:Rodrigues2" \end_inset . \end_layout \begin_layout Subsection The Adjoint Map \end_layout \begin_layout Standard For rotation matrices \begin_inset Formula $R$ \end_inset we can prove the following identity (see \begin_inset CommandInset ref LatexCommand vref reference "proof1" \end_inset ): \begin_inset Formula \begin{equation} R\Skew{\omega}R^{T}=\Skew{R\omega}\label{eq:property1} \end{equation} \end_inset Hence, given property \begin_inset CommandInset ref LatexCommand eqref reference "eq:property1" \end_inset , the adjoint map for \begin_inset Formula $\sothree$ \end_inset simplifies to \begin_inset Formula \[ \Ad R{\Skew{\omega}}=R\Skew{\omega}R^{T}=\Skew{R\omega} \] \end_inset and this can be expressed in exponential coordinates simply by rotating the axis \begin_inset Formula $\omega$ \end_inset to \begin_inset Formula $R\omega$ \end_inset . \end_layout \begin_layout Standard As an example, to apply an axis-angle rotation \begin_inset Formula $\omega$ \end_inset to a point \begin_inset Formula $p$ \end_inset in the frame \begin_inset Formula $R$ \end_inset , we could: \end_layout \begin_layout Enumerate First transform \begin_inset Formula $p$ \end_inset back to the world frame, apply \begin_inset Formula $\omega$ \end_inset , and then rotate back: \begin_inset Formula \[ q=Re^{\Skew{\omega}}R^{T}p \] \end_inset \end_layout \begin_layout Enumerate Immediately apply the transformed axis-angle transformation \begin_inset Formula $\Ad R{\Skew{\omega}}=\Skew{R\omega}$ \end_inset : \begin_inset Formula \[ q=e^{\Skew{R\omega}}p \] \end_inset \end_layout \begin_layout Subsection Actions \end_layout \begin_layout Standard In the case of \begin_inset Formula $\SOthree$ \end_inset the vector space is \begin_inset Formula $\Rthree$ \end_inset , and the group action corresponds to rotating a point \begin_inset Formula \[ q=Rp \] \end_inset We would now like to know what an incremental rotation parameterized by \begin_inset Formula $\omega$ \end_inset would do: \begin_inset Formula \[ q(\omega)=Re^{\Skew{\omega}}p \] \end_inset hence the derivative is: \begin_inset Formula \[ \deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)=R\deriv{}{\omega}\left(\Skew{\omega}p\right)=R\Skew{-p} \] \end_inset To show the last equality note that \begin_inset Formula \[ \Skew{\omega}p=\omega\times p=-p\times\omega=\Skew{-p}\omega \] \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Section 3D Rigid Transformations \end_layout \begin_layout Standard The Lie group \begin_inset Formula $\SEthree$ \end_inset is a subgroup of the general linear group \begin_inset Formula $GL(4)$ \end_inset of \begin_inset Formula $4\times4$ \end_inset invertible matrices of the form \begin_inset Formula \[ T\define\left[\begin{array}{cc} R & t\\ 0 & 1 \end{array}\right] \] \end_inset where \begin_inset Formula $R\in\SOthree$ \end_inset is a rotation matrix and \begin_inset Formula $t\in\Rthree$ \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 is the vector space of \begin_inset Formula $4\times4$ \end_inset twists \begin_inset Formula $\xihat$ \end_inset parameterized by the \emph on twist coordinates \emph default \begin_inset Formula $\xi\in\Rsix$ \end_inset , with the mapping \begin_inset CommandInset citation LatexCommand cite key "Murray94book" \end_inset \begin_inset Formula \[ \xi\define\left[\begin{array}{c} \omega\\ v \end{array}\right]\rightarrow\xihat\define\left[\begin{array}{cc} \Skew{\omega} & v\\ 0 & 0 \end{array}\right] \] \end_inset Note we follow Frank Park's convention and reserve the first three components for rotation, and the last three for translation. Hence, with this parameterization, the generators for \begin_inset Formula $\SEthree$ \end_inset are \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 \[ G^{4}=\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right)\mbox{}G^{5}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right)\mbox{ }G^{6}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{array}\right) \] \end_inset Applying the exponential map to a twist \begin_inset Formula $\xi$ \end_inset yields a screw motion yielding an element in \begin_inset Formula $\SEthree$ \end_inset : \begin_inset Formula \[ T=\exp\xihat \] \end_inset A closed form solution for the exponential map is given in \begin_inset CommandInset citation LatexCommand cite after "page 42" key "Murray94book" \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 \[ \exp\left(\widehat{\left[\begin{array}{c} \omega\\ v \end{array}\right]}t\right)=\left[\begin{array}{cc} e^{\Skew{\omega}t} & (I-e^{\Skew{\omega}t})\left(\omega\times v\right)+\omega\omega^{T}vt\\ 0 & 1 \end{array}\right] \] \end_inset \end_layout \begin_layout Subsection The Adjoint Map \end_layout \begin_layout Standard The adjoint is \begin_inset Formula \begin{eqnarray*} \Ad T{\xihat} & = & T\xihat T^{-1}\\ & = & \left[\begin{array}{cc} R & t\\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} \Skew{\omega} & v\\ 0 & 0 \end{array}\right]\left[\begin{array}{cc} R^{T} & -R^{T}t\\ 0 & 1 \end{array}\right]\\ & = & \left[\begin{array}{cc} \Skew{R\omega} & -\Skew{R\omega}t+Rv\\ 0 & 0 \end{array}\right]\\ & = & \left[\begin{array}{cc} \Skew{R\omega} & t\times R\omega+Rv\\ 0 & 0 \end{array}\right] \end{eqnarray*} \end_inset From this we can express the Adjoint map in terms of twist coordinates (see also \begin_inset CommandInset citation LatexCommand cite key "Murray94book" \end_inset and FP): \begin_inset Formula \[ \left[\begin{array}{c} \omega'\\ v' \end{array}\right]=\left[\begin{array}{cc} R & 0\\ \Skew tR & R \end{array}\right]\left[\begin{array}{c} \omega\\ v \end{array}\right] \] \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 \[ \hat{q}=\left[\begin{array}{c} q\\ 1 \end{array}\right]=\left[\begin{array}{c} Rp+t\\ 1 \end{array}\right]=\left[\begin{array}{cc} R & t\\ 0 & 1 \end{array}\right]\left[\begin{array}{c} p\\ 1 \end{array}\right]=T\hat{p} \] \end_inset We would now like to know what an incremental pose parameterized by \begin_inset Formula $\xi$ \end_inset would do: \begin_inset Formula \[ \hat{q}(\xi)=Te^{\xihat}\hat{p} \] \end_inset hence the derivative is \begin_inset Formula \[ \deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right) \] \end_inset where \begin_inset Formula $\xihat\hat{p}$ \end_inset corresponds to a velocity in \begin_inset Formula $\mathbb{R}^{4}$ \end_inset (in the local \begin_inset Formula $T$ \end_inset frame): \begin_inset Formula \[ \xihat\hat{p}=\left[\begin{array}{cc} \Skew{\omega} & v\\ 0 & 0 \end{array}\right]\left[\begin{array}{c} p\\ 1 \end{array}\right]=\left[\begin{array}{c} \omega\times p+v\\ 0 \end{array}\right] \] \end_inset Notice how velocities are analogous to points at infinity in projective geometry: they correspond to free vectors indicating a direction and magnitude of change. \end_layout \begin_layout Standard By only taking the top three rows, we can write this as a velocity in \begin_inset Formula $\Rthree$ \end_inset , as the product of a \begin_inset Formula $3\times6$ \end_inset matrix \begin_inset Formula $H_{p}$ \end_inset that acts upon the exponential coordinates \begin_inset Formula $\xi$ \end_inset directly: \begin_inset Formula \[ \omega\times p+v=-p\times\omega+v=\left[\begin{array}{cc} -\Skew p & I_{3}\end{array}\right]\left[\begin{array}{c} \omega\\ v \end{array}\right] \] \end_inset yielding the derivative \begin_inset Formula \[ \deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right)=T\left[\begin{array}{cc} -\Skew p & I_{3}\\ 0 & 0 \end{array}\right] \] \end_inset The inverse action \begin_inset Formula $T^{-1}p$ \end_inset is \begin_inset Formula \[ \hat{q}=\left[\begin{array}{c} q\\ 1 \end{array}\right]=\left[\begin{array}{c} R^{T}(p-t)\\ 1 \end{array}\right]=\left[\begin{array}{cc} R^{T} & -R^{T}t\\ 0 & 1 \end{array}\right]\left[\begin{array}{c} p\\ 1 \end{array}\right]=T^{-1}\hat{p} \] \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \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 pagebreak \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 hierarchy among the associated Lie algebras. 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 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 \[ e^{xG_{5}}=\exp\left[\begin{array}{ccc} x & 0 & 0\\ 0 & -x & 0\\ 0 & 0 & 0 \end{array}\right]=\left[\begin{array}{ccc} e^{x} & 0 & 0\\ 0 & 1/e^{x} & 0\\ 0 & 0 & 1 \end{array}\right] \] \end_inset \begin_inset Formula \[ e^{xG_{6}}=\exp\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & -x & 0\\ 0 & 0 & x \end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1/e^{x} & 0\\ 0 & 0 & e^{x} \end{array}\right] \] \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 \[ e^{xG_{5}}=\exp\left[\begin{array}{ccc} x & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -x \end{array}\right]=\left[\begin{array}{ccc} e^{x} & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1/e^{x} \end{array}\right] \] \end_inset \begin_inset Formula \[ e^{xG_{6}}=\exp\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & x & 0\\ 0 & 0 & -x \end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & e^{x} & 0\\ 0 & 0 & 1/e^{x} \end{array}\right] \] \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 Standard \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 "proof1" \end_inset \end_layout \begin_layout Standard We can prove the following identity for rotation matrices \begin_inset Formula $R$ \end_inset , \begin_inset Formula \begin{eqnarray} R\Skew{\omega}R^{T} & = & R\Skew{\omega}\left[\begin{array}{ccc} a_{1} & a_{2} & a_{3}\end{array}\right]\nonumber \\ & = & R\left[\begin{array}{ccc} \omega\times a_{1} & \omega\times a_{2} & \omega\times a_{3}\end{array}\right]\nonumber \\ & = & \left[\begin{array}{ccc} a_{1}(\omega\times a_{1}) & a_{1}(\omega\times a_{2}) & a_{1}(\omega\times a_{3})\\ a_{2}(\omega\times a_{1}) & a_{2}(\omega\times a_{2}) & a_{2}(\omega\times a_{3})\\ a_{3}(\omega\times a_{1}) & a_{3}(\omega\times a_{2}) & a_{3}(\omega\times a_{3}) \end{array}\right]\nonumber \\ & = & \left[\begin{array}{ccc} \omega(a_{1}\times a_{1}) & \omega(a_{2}\times a_{1}) & \omega(a_{3}\times a_{1})\\ \omega(a_{1}\times a_{2}) & \omega(a_{2}\times a_{2}) & \omega(a_{3}\times a_{2})\\ \omega(a_{1}\times a_{3}) & \omega(a_{2}\times a_{3}) & \omega(a_{3}\times a_{3}) \end{array}\right]\nonumber \\ & = & \left[\begin{array}{ccc} 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{proof1} \end{eqnarray} \end_inset where \begin_inset Formula $a_{1}$ \end_inset , \begin_inset Formula $a_{2}$ \end_inset , and \begin_inset Formula $a_{3}$ \end_inset are the \emph on rows \emph default of \begin_inset Formula $R$ \end_inset . Above we made use of the orthogonality of rotation matrices and the triple product rule: \begin_inset Formula \[ a(b\times c)=b(c\times a)=c(a\times b) \] \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 \begin_layout Standard \begin_inset CommandInset bibtex LatexCommand bibtex bibfiles "../../../papers/refs" options "plain" \end_inset \end_layout \end_body \end_document