diff --git a/doc/LieGroups.lyx b/doc/LieGroups.lyx index cf7ff7c9e..a2ab4c12e 100644 --- a/doc/LieGroups.lyx +++ b/doc/LieGroups.lyx @@ -123,6 +123,12 @@ Lie Groups \end_inset +\begin_inset FormulaMacro +\newcommand{\AAdd}[1]{\mathbf{\mathop{Ad}}{}_{#1}} +{\mathbf{\mathop{Ad}}{}_{#1}} +\end_inset + + \end_layout \begin_layout Standard @@ -583,7 +589,7 @@ T_{1}T_{2}=(x_{1},\, y_{1},\,\theta_{1})(x_{2},\, y_{2},\,\theta_{2})=(x_{1}+\co \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 @@ -697,7 +703,7 @@ e^{x}=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=\sum_{k=0}^{\infty \end_inset -The series can be similarly defined for square matrices,and the final result +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 $\xi=(v,\omega)$ @@ -742,8 +748,9 @@ special Euclidean group \end_inset . - It is called a Lie group because it is both a manifold and a group, and - its group operation is smooth when operating on this manifold. + 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 $\setwo$ @@ -778,7 +785,7 @@ A Lie group \begin_inset Formula $G$ \end_inset - is a manifold that possesses a smooth group operation. + is a a group and a manifold that possesses a smooth group operation. Associated with it is a Lie Algebra \begin_inset Formula $\gg$ \end_inset @@ -896,8 +903,15 @@ Instead, \begin_inset Formula $Z$ \end_inset - can be calculated using the Baker-Campbell-Hausdorff (BCH) formula: -\begin_inset Foot + 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 @@ -906,7 +920,7 @@ http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula \end_inset - +: \begin_inset Formula \[ Z=X+Y+[X,Y]/2+[X-Y,[X,Y]]/12-[Y,[X,[X,Y]]]/24+\ldots\] @@ -929,7 +943,7 @@ For \begin_inset Formula $n$ \end_inset --dimensional matrix Lie groups, the Lie algebra +-dimensional matrix Lie groups, as a vector space the Lie algebra \begin_inset Formula $\gg$ \end_inset @@ -937,7 +951,7 @@ For \begin_inset Formula $\mathbb{R}^{n}$ \end_inset -, and we can define the wedge operator +, and we can define the hat operator \begin_inset CommandInset citation LatexCommand cite after "page 41" @@ -990,7 +1004,7 @@ where the \begin_inset Formula $n\times n$ \end_inset - matrices known as the Lie group generators. + matrices known as Lie group generators. The meaning of the map \begin_inset Formula $x\rightarrow\xhat$ \end_inset @@ -1003,12 +1017,92 @@ where the \end_layout \begin_layout Subsection -The Adjoint Map +Tangent Spaces and the Tangent Bundle \end_layout \begin_layout Standard -Below we frequently make use of the equality -\begin_inset Foot +The tangent space +\begin_inset Formula $T_{g}G$ +\end_inset + + at a group element +\begin_inset Formula $g\in G$ +\end_inset + + is the vector space of tangent vectors at +\begin_inset Formula $g$ +\end_inset + +. + The tangent bundle +\begin_inset Formula $TG$ +\end_inset + + is the set of all tangent vectors +\begin_inset Formula \[ +TG\define\bigcup_{g\in G}T_{g}G\] + +\end_inset + +A vector field +\begin_inset Formula $X:G\rightarrow TG$ +\end_inset + + assigns to each element +\begin_inset Formula $g$ +\end_inset + + as single tangent vector +\begin_inset Formula $x\in T_{g}G$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +The Adjoint Map and Adjoint Representation +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +We do not fully understand this yet!!! +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The +\emph on +adjoint map +\emph default + +\begin_inset Formula $\AAdd g$ +\end_inset + + maps a group element +\begin_inset Formula $a\in G$ +\end_inset + + to its conjugate +\begin_inset Formula $gag^{-1}$ +\end_inset + + by +\begin_inset Formula $g$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Below we frequently make use of the equality +\begin_inset Note Note status collapsed \begin_layout Plain Layout @@ -1019,7 +1113,7 @@ http://en.wikipedia.org/wiki/Exponential_map \begin_inset Formula \[ -ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\] +\AAdd ge^{\xhat}=ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\] \end_inset @@ -1031,8 +1125,17 @@ where \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 Note Note +status open + +\begin_layout Plain Layout +The intuitive explanation is that a change \begin_inset Formula $\exp\left(\xhat\right)$ \end_inset @@ -1049,8 +1152,16 @@ where \end_inset . - In the case of a matrix group the ajoint can be written as -\begin_inset Foot +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +In the case of a matrix group the adjoint can be written as +\begin_inset Note Note status collapsed \begin_layout Plain Layout @@ -1307,13 +1418,9 @@ all \end_inset -which we can write in terms of -\begin_inset Formula $\omega$ -\end_inset - - as +i.e., \begin_inset Formula \[ -\Ad R\omega=\omega\] +\Ad R\what=\what\] \end_inset @@ -2292,6 +2399,8 @@ The action of \begin_inset Formula \[ \hat{q}=\left[\begin{array}{c} q\\ +1\end{array}\right]=\left[\begin{array}{c} +Rp+t\\ 1\end{array}\right]=\left[\begin{array}{cc} R & t\\ 0 & 1\end{array}\right]\left[\begin{array}{c} @@ -2372,6 +2481,24 @@ v\end{array}\right]\] \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 diff --git a/doc/LieGroups.pdf b/doc/LieGroups.pdf index 005b8c835..1aa0c934e 100644 Binary files a/doc/LieGroups.pdf and b/doc/LieGroups.pdf differ