gtsam/doc/LieGroups.lyx

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\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