diff --git a/doc/LieGroups.lyx b/doc/LieGroups.lyx index 1df1c4a60..319ded054 100644 --- a/doc/LieGroups.lyx +++ b/doc/LieGroups.lyx @@ -419,9 +419,9 @@ The solution to this trivial differential equation is, with \begin_inset Formula $x$ \end_inset --position f the robot, +-position of the robot, \begin_inset Formula \[ -x=x_{0}+v_{x}t\] +x_{t}=x_{0}+v_{x}t\] \end_inset @@ -431,13 +431,13 @@ A similar story holds for translation in the direction, and in fact for translations in general: \begin_inset Formula \[ -(x,\, y,\,\theta)=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0})\] +(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 \[ -(x,\, y,\,\theta)=(x_{0},\, y_{0},\,\theta_{0}+\omega t)\] +(x_{t},\, y_{t},\,\theta_{t})=(x_{0},\, y_{0},\,\theta_{0}+\omega t)\] \end_inset @@ -488,7 +488,7 @@ Robot moving along a circular trajectory. However, if we combine translation and rotation, the story breaks down! We cannot write \begin_inset Formula \[ -(x,\, y,\,\theta)=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t)\] +(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t)\] \end_inset @@ -499,7 +499,7 @@ The reason is that, if we move the robot a tiny bit according to the velocity , we have (to first order) \begin_inset Formula \[ -(x_{t+\delta},\, y_{t+\delta},\,\theta_{t+\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)\] +(x_{\delta},\, y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)\] \end_inset @@ -530,6 +530,43 @@ If rotation and translation commuted, we could do all rotations before leaving \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 /Users/dellaert/borg/gtsam/doc/images/n-steps.pdf + +\end_inset + + +\begin_inset Caption + +\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 $T_{1}$ @@ -574,52 +611,7 @@ R=\left[\begin{array}{cc} \end_inset - -\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 - -\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 - - -\end_layout - -\begin_layout Standard -Now, a +Now a \begin_inset Quotes eld \end_inset @@ -638,7 +630,7 @@ T(\delta)=\left[\begin{array}{ccc} 0 & 0 & 1\end{array}\right]=I+\delta\left[\begin{array}{ccc} 0 & -\omega & v_{x}\\ \omega & 0 & v_{y}\\ -0 & 0 & 1\end{array}\right]\] +0 & 0 & 0\end{array}\right]\] \end_inset @@ -655,7 +647,7 @@ Let us define the \xihat\define\left[\begin{array}{ccc} 0 & -\omega & v_{x}\\ \omega & 0 & v_{y}\\ -0 & 0 & 1\end{array}\right]\] +0 & 0 & 0\end{array}\right]\] \end_inset @@ -715,13 +707,17 @@ The series can be similarly defined for square matrices,and the final result \begin_inset Formula $ $ \end_inset - as the matrix exponential of + as the +\emph on +matrix exponential +\emph default + of \begin_inset Formula $\xihat$ \end_inset : \begin_inset Formula \[ -T(t)=e^{t\xihat}\define\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}=\sum_{k=0}^{\infty}\frac{\left(t\xihat\right)^{k}}{k!}\] +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 @@ -746,8 +742,8 @@ special Euclidean group \end_inset . - It is called a Lie group because it is both a manifold, and its group operation - is smooth when operating on this manifold. + 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. The space of 2D twists, together with a special binary operation to be defined below, is called the Lie algebra \begin_inset Formula $\setwo$ @@ -813,7 +809,7 @@ logarithm \begin_inset Quotes erd \end_inset -: + \begin_inset Formula \[ \log:G\rightarrow\gg\] @@ -855,8 +851,11 @@ The Lie Algebra \end_inset . - The relationship with the group operation is as follows: for commutative - Lie groups vector addition +\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 @@ -1460,7 +1459,7 @@ T=\left[\begin{array}{cc} 0 & t\\ 0 & 1\end{array}\right]\left[\begin{array}{cc} R & 0\\ -0 & k\end{array}\right]\] +0 & 1\end{array}\right]\] \end_inset @@ -1770,18 +1769,11 @@ where . Hence, a slightly more efficient variant is -\end_layout - -\begin_layout Standard \begin_inset Formula \[ e^{\what}=\cos\theta I+\what sin\theta+\omega\omega^{T}(1\text{−}cos\theta)\] \end_inset - -\end_layout - -\begin_layout Standard Since \begin_inset Formula $\SOthree$ \end_inset @@ -1935,7 +1927,7 @@ R\Skew{\omega}R^{T}=\Skew{R\omega}\label{eq:property1}\end{equation} Hence, given property \begin_inset CommandInset ref LatexCommand eqref -reference "proof1" +reference "eq:property1" \end_inset @@ -2521,7 +2513,8 @@ From this it can be gleaned that the groups \bar no \noun off \color none -By choosing the generators carefully we maintain this subgroup hierarchy. +By choosing the generators carefully we maintain this hierarchy among the + associated Lie algebras. In particular, \begin_inset Formula $\setwo$ \end_inset @@ -2612,10 +2605,6 @@ a_{4}+a_{3} & -a_{5}-a_{6} & a_{2}\\ \end_inset - -\end_layout - -\begin_layout Standard Note that \begin_inset Formula $G_{5}$ \end_inset @@ -2634,10 +2623,10 @@ Note that but without changing the determinant: \begin_inset Formula \[ -e^{xG_{5}}=\exp\left(\left[\begin{array}{ccc} +e^{xG_{5}}=\exp\left[\begin{array}{ccc} x & 0 & 0\\ 0 & -x & 0\\ -0 & 0 & 0\end{array}\right]\right)=\left[\begin{array}{ccc} +0 & 0 & 0\end{array}\right]=\left[\begin{array}{ccc} e^{x} & 0 & 0\\ 0 & 1/e^{x} & 0\\ 0 & 0 & 1\end{array}\right]\] @@ -2646,10 +2635,10 @@ e^{x} & 0 & 0\\ \begin_inset Formula \[ -e^{xG_{6}}=\exp\left(\left[\begin{array}{ccc} +e^{xG_{6}}=\exp\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & -x & 0\\ -0 & 0 & x\end{array}\right]\right)=\left[\begin{array}{ccc} +0 & 0 & x\end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1/e^{x} & 0\\ 0 & 0 & e^{x}\end{array}\right]\] @@ -2696,10 +2685,10 @@ and hence \color inherit \begin_inset Formula \[ -e^{xG_{5}}=\exp\left(\left[\begin{array}{ccc} +e^{xG_{5}}=\exp\left[\begin{array}{ccc} x & 0 & 0\\ 0 & 0 & 0\\ -0 & 0 & -x\end{array}\right]\right)=\left[\begin{array}{ccc} +0 & 0 & -x\end{array}\right]=\left[\begin{array}{ccc} e^{x} & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1/e^{x}\end{array}\right]\] @@ -2708,10 +2697,10 @@ e^{x} & 0 & 0\\ \begin_inset Formula \[ -e^{xG_{6}}=\exp\left(\left[\begin{array}{ccc} +e^{xG_{6}}=\exp\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & x & 0\\ -0 & 0 & -x\end{array}\right]\right)=\left[\begin{array}{ccc} +0 & 0 & -x\end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & e^{x} & 0\\ 0 & 0 & 1/e^{x}\end{array}\right]\]