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gtsam-coordinate-frames.lyx
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
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\begin_layout Section
Introduction
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This document describes the coordinate frame conventions in which GTSAM
inputs and represents states and uncertainties.
When specifying initial conditions, measurements and their uncertainties,
and interpreting estimated uncertainties and the results of geometry operations
, the coordinate frame convention comes into play.
\end_layout
\begin_layout Standard
GTSAM as consistently as possible represents all states and uncertainties
in the body frame.
In cases where several frames are used simultaneously, a good rule of thumb
is that measurements and uncertainties will be represented in the
\begin_inset Quotes eld
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last
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frame of the series.
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\begin_layout Section
Frame Conventions in Geometry, Lie Group, and Manifold Operations
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in
\begin_inset Formula $g$
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset Formula $p$
\end_inset
in
\begin_inset Formula $a$
\end_inset
\end_layout
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</cell>
<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $q=p$
\end_inset
in
\begin_inset Formula $g$
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset Formula $q=ap$
\end_inset
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</cell>
<cell alignment="left" valignment="top" topline="true" bottomline="true" usebox="none">
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\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:Coordinate-frame-transformations"
\end_inset
Coordinate frame transformations performed by GTSAM geometry operations.
Here,
\begin_inset Formula $a$
\end_inset
,
\begin_inset Formula $b$
\end_inset
,
\begin_inset Formula $c$
\end_inset
, and
\begin_inset Formula $g$
\end_inset
are Lie group elements (Pose2, Pose3, Rot2, Rot3, Point2, Point3,
\emph on
etc
\emph default
).
\begin_inset Formula $\delta$
\end_inset
is a set of Lie algebra coordinates (i.e.
linear update, linear delta, tangent space coordinates), and
\begin_inset Formula $\mathrm{X}$
\end_inset
is a Lie group type (e.g.
Pose2).
\begin_inset Formula $p$
\end_inset
and
\begin_inset Formula $q$
\end_inset
are the objects of Lie group actions (Point2, Point3,
\emph on
etc
\emph default
).
\end_layout
\end_inset
\end_layout
\end_inset
At the core of most coordinate frame usage in GTSAM are geometry and Lie
group operations.
We explain the geometry and Lie group operations in GTSAM in terms of the
coordinate frame transformations they perform, detailed in Table
\begin_inset space ~
\end_inset
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Coordinate-frame-transformations"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
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\align center
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\begin_inset Tabular
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<column alignment="left" valignment="top" width="1cm">
<column alignment="left" valignment="top" width="1.25cm">
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<row>
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