#LyX 1.6.5 created this file. For more info see http://www.lyx.org/ \lyxformat 345 \begin_document \begin_header \textclass article \use_default_options false \begin_modules theorems-std \end_modules \language english \inputencoding auto \font_roman times \font_sans default \font_typewriter default \font_default_family rmdefault \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \paperfontsize 12 \spacing single \use_hyperref false \papersize default \use_geometry true \use_amsmath 1 \use_esint 0 \cite_engine basic \use_bibtopic false \paperorientation portrait \leftmargin 1in \topmargin 1in \rightmargin 1in \bottommargin 1in \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \author "" \author "" \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 Basic Lie Group Concepts \end_layout \begin_layout Subsection A Manifold and a Group \end_layout \begin_layout Standard A Lie group \begin_inset Formula $G$ \end_inset is a manifold that possesses a smooth group operation. Associated with it is a Lie Algebra \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 exponential map \begin_inset Formula \[ \exp:\gg\rightarrow G\] \end_inset and a corresponding inverse \begin_inset Formula \[ \log:G\rightarrow\gg\] \end_inset that maps elements in G to an element in \begin_inset Formula $\gg$ \end_inset . \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 Lie bracket \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, in matrix Lie groups, the Lie bracket is given by \begin_inset Formula $[A,B]\define AB-BA$ \end_inset . The relationship with 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 Foot 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, 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 mapping \begin_inset Formula \[ \hat{}:\mathbb{R}^{n}\rightarrow\gg\] \end_inset \begin_inset Formula \[ \hat{}:x\rightarrow\xhat\] \end_inset which maps \begin_inset Formula $n$ \end_inset -vectors \begin_inset Formula $x\in$ \end_inset \begin_inset Formula $\Rn$ \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 \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 the 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 be very intuitive. \end_layout \begin_layout Subsection The Adjoint Map \end_layout \begin_layout Standard Below we frequently make use of the equality \begin_inset Foot status collapsed \begin_layout Plain Layout http://en.wikipedia.org/wiki/Exponential_map \end_layout \end_inset \begin_inset Formula \[ ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\] \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 . The intuitive explanation is that a change \begin_inset Formula $\exp\left(\xhat\right)$ \end_inset defined around the orgin, 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 . In the case of a matrix group the ajoint can be written as \begin_inset Foot status collapsed \begin_layout Plain Layout http://en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_group \end_layout \end_inset \begin_inset Formula \[ \Ad T{\xhat}\define Te^{\xhat}T^{-1}\] \end_inset and hence we have \end_layout \begin_layout Standard \begin_inset Formula \[ Te^{\xhat}T^{-1}=e^{T\xhat T^{-1}}\] \end_inset where both \begin_inset Formula $T$ \end_inset and \begin_inset Formula $\xhat$ \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 Subsection Actions \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 \[ q=Tp\] \end_inset with \begin_inset Formula $p,q\in\mathbb{R}^{n}$ \end_inset and \begin_inset Formula $T\in GL(n)$ \end_inset . \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 \[ \hat{}:\theta\rightarrow\that=\skew{\theta}\] \end_inset which maps the angle \begin_inset Formula $\theta$ \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 $\skew{\theta}$ \end_inset : \family default \series default \shape default \size default \emph default \bar default \noun default \color inherit \begin_inset Formula \[ \skew{\theta}=\left[\begin{array}{cc} 0 & -\theta\\ \theta & 0\end{array}\right]\] \end_inset The exponential map can be computed in closed form as \begin_inset Formula \[ R=e^{\skew{\theta}}=\left[\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{array}\right]\] \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 $\theta$ \end_inset would do: \begin_inset Formula \[ q(\text{\theta})=Re^{\skew{\theta}}p\] \end_inset hence the derivative is: \begin_inset Formula \[ \deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\skew{\theta}}p\right)=R\deriv{}{\omega}\left(\skew{\theta}p\right)=RH_{p}\] \end_inset Note that \begin_inset Formula \begin{equation} \skew{\theta}\left[\begin{array}{c} x\\ y\end{array}\right]=\theta R_{\pi/2}\left[\begin{array}{c} x\\ y\end{array}\right]=\theta\left[\begin{array}{c} -y\\ x\end{array}\right]\label{eq:RestrictedCross}\end{equation} \end_inset which acts like a restricted \begin_inset Quotes eld \end_inset cross product \begin_inset Quotes erd \end_inset in the plane. \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. Its 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 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 \[ T=\exp\xihat\] \end_inset A closed form solution for the exponential map is in the works... \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-\omega R_{\pi/2}t\\ 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 \[ \left[\begin{array}{c} v'\\ \omega'\end{array}\right]=\left[\begin{array}{cc} R & -R_{\pi/2}t\\ 0 & 1\end{array}\right]\left[\begin{array}{c} v\\ \omega\end{array}\right]\] \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 , 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. The exponential map can be computed in closed form using Rodrigues' formula. \end_layout \begin_layout Standard 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 where the \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 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 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 and this is defines the 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/\left\Vert \omega\right\Vert $ \end_inset defines the rotation axis, and its magnitude the amount of rotation \begin_inset Formula $\theta$ \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 "remove" \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 "remove" \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 \[ q=Re^{\Skew{\omega}}R^{T}\] \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 \[ \deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)=R\deriv{}{\omega}\left(\Skew{\omega}p\right)=RH_{p}\] \end_inset To calculate \begin_inset Formula $H_{p}$ \end_inset we make use of \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. 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 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 \[ \hat{q}=\left[\begin{array}{c} q\\ 1\end{array}\right]=\left[\begin{array}{cc} R & t\\ 0 & 1\end{array}\right]\left[\begin{array}{c} p\\ 1\end{array}\right]=T\hat{p}\] \end_inset We would now like to know what an incremental rotation 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 (following the exposition in Section \begin_inset CommandInset ref LatexCommand ref reference "sec:Derivatives-of-Actions" \end_inset ): \begin_inset Formula \[ \deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right)=TH_{p}\] \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 anologous 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 \[ \omega\times p+v=-p\times\omega+v=\left[\begin{array}{cc} -\Skew p & I_{3}\end{array}\right]\left[\begin{array}{c} \omega\\ v\end{array}\right]=H_{p}\xi\] \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" \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{remove}\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 \end_layout \begin_layout Standard \begin_inset CommandInset bibtex LatexCommand bibtex bibfiles "/Users/dellaert/papers/refs" options "plain" \end_inset \end_layout \end_body \end_document