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							|  | @ -55,10 +55,249 @@ Frank Dellaert | |||
| \end_layout | ||||
| 
 | ||||
| \begin_layout Standard | ||||
| \begin_inset CommandInset include | ||||
| LatexCommand include | ||||
| filename "macros.lyx" | ||||
| \begin_inset Note Comment | ||||
| status open | ||||
| 
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| \begin_layout Plain Layout | ||||
| Derivatives | ||||
| \end_layout | ||||
| 
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| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\deriv}[2]{\frac{\partial#1}{\partial#2}} | ||||
| {\frac{\partial#1}{\partial#2}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\at}[2]{#1\biggr\rvert_{#2}} | ||||
| {#1\biggr\rvert_{#2}} | ||||
| \end_inset | ||||
| 
 | ||||
|   | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\Jac}[3]{ \at{\deriv{#1}{#2}} {#3} } | ||||
| {\at{\deriv{#1}{#2}}{#3}} | ||||
| \end_inset | ||||
| 
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| 
 | ||||
| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset Note Comment | ||||
| status open | ||||
| 
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| \begin_layout Plain Layout | ||||
| Lie Groups | ||||
| \end_layout | ||||
| 
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| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\xhat}{\hat{x}} | ||||
| {\hat{x}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
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| \begin_inset FormulaMacro | ||||
| \newcommand{\yhat}{\hat{y}} | ||||
| {\hat{y}} | ||||
| \end_inset | ||||
| 
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| 
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| \begin_inset FormulaMacro | ||||
| \newcommand{\Ad}[1]{Ad_{#1}} | ||||
| {Ad_{#1}} | ||||
| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\define}{\stackrel{\Delta}{=}} | ||||
| {\stackrel{\Delta}{=}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
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| \begin_inset FormulaMacro | ||||
| \newcommand{\gg}{\mathfrak{g}} | ||||
| {\mathfrak{g}} | ||||
| \end_inset | ||||
| 
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| 
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| \begin_inset FormulaMacro | ||||
| \newcommand{\Rn}{\mathbb{R}^{n}} | ||||
| {\mathbb{R}^{n}} | ||||
| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset Note Comment | ||||
| status open | ||||
| 
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| \begin_layout Plain Layout | ||||
| SO(2) | ||||
| \end_layout | ||||
| 
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| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}} | ||||
| {\mathfrak{\mathbb{R}^{2}}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
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| \begin_inset FormulaMacro | ||||
| \newcommand{\SOtwo}{SO(2)} | ||||
| {SO(2)} | ||||
| \end_inset | ||||
| 
 | ||||
| 
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| \begin_inset FormulaMacro | ||||
| \newcommand{\sotwo}{\mathfrak{so(2)}} | ||||
| {\mathfrak{so(2)}} | ||||
| \end_inset | ||||
| 
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| 
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| \begin_inset FormulaMacro | ||||
| \newcommand{\that}{\hat{\theta}} | ||||
| {\hat{\theta}} | ||||
| \end_inset | ||||
| 
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| 
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| \begin_inset FormulaMacro | ||||
| \newcommand{\skew}[1]{[#1]_{+}} | ||||
| {[#1]_{+}} | ||||
| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset Note Comment | ||||
| status open | ||||
| 
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| \begin_layout Plain Layout | ||||
| SE(2) | ||||
| \end_layout | ||||
| 
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| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\SEtwo}{SE(2)} | ||||
| {SE(2)} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\setwo}{\mathfrak{se(2)}} | ||||
| {\mathfrak{se(2)}} | ||||
| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset Note Comment | ||||
| status open | ||||
| 
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| \begin_layout Plain Layout | ||||
| SO(3) | ||||
| \end_layout | ||||
| 
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| \end_inset | ||||
| 
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| 
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| \end_layout | ||||
| 
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| \begin_layout Standard | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\Rthree}{\mathfrak{\mathbb{R}^{3}}} | ||||
| {\mathfrak{\mathbb{R}^{3}}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\SOthree}{SO(3)} | ||||
| {SO(3)} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\sothree}{\mathfrak{so(3)}} | ||||
| {\mathfrak{so(3)}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\what}{\hat{\omega}} | ||||
| {\hat{\omega}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\Skew}[1]{[#1]_{\times}} | ||||
| {[#1]_{\times}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \end_layout | ||||
| 
 | ||||
| \begin_layout Standard | ||||
| \begin_inset Note Comment | ||||
| status open | ||||
| 
 | ||||
| \begin_layout Plain Layout | ||||
| SE(3) | ||||
| \end_layout | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \end_layout | ||||
| 
 | ||||
| \begin_layout Standard | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\Rsix}{\mathfrak{\mathbb{R}^{6}}} | ||||
| {\mathfrak{\mathbb{R}^{6}}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\SEthree}{SE(3)} | ||||
| {SE(3)} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\sethree}{\mathfrak{se(3)}} | ||||
| {\mathfrak{se(3)}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
| \begin_inset FormulaMacro | ||||
| \newcommand{\xihat}{\hat{\xi}} | ||||
| {\hat{\xi}} | ||||
| \end_inset | ||||
| 
 | ||||
| 
 | ||||
|  | @ -289,6 +528,72 @@ Hence, now we undo | |||
|  frame. | ||||
| \end_layout | ||||
| 
 | ||||
| \begin_layout Subsection* | ||||
| Numerical Derivatives | ||||
| \end_layout | ||||
| 
 | ||||
| \begin_layout Standard | ||||
| Let's examine | ||||
| \begin_inset Formula \[ | ||||
| f\left(g\right)e^{\yhat}=f\left(ge^{\xhat}\right)\] | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
| and multiply with  | ||||
| \begin_inset Formula $f(g)^{-1}$ | ||||
| \end_inset | ||||
| 
 | ||||
|  on both sides then take the log (which in our case returns  | ||||
| \begin_inset Formula $y$ | ||||
| \end_inset | ||||
| 
 | ||||
| , not  | ||||
| \begin_inset Formula $\yhat$ | ||||
| \end_inset | ||||
| 
 | ||||
| ): | ||||
| \begin_inset Formula \[ | ||||
| y(x)=\log\left[f\left(g\right)^{-1}f\left(ge^{\xhat}\right)\right]\] | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
| Let us look at  | ||||
| \begin_inset Formula $x=0$ | ||||
| \end_inset | ||||
| 
 | ||||
| , and perturb in direction  | ||||
| \begin_inset Formula $i$ | ||||
| \end_inset | ||||
| 
 | ||||
| ,  | ||||
| \begin_inset Formula $e_{i}=[0,0,d,0,0]$ | ||||
| \end_inset | ||||
| 
 | ||||
| . | ||||
|  Then take derivative,  | ||||
| \begin_inset Formula \[ | ||||
| \deriv{y(d)}d\define\lim_{d->0}\frac{y(d)-y(0)}{d}=\lim_{d->0}\frac{1}{d}\log\left[f\left(g\right)^{-1}f\left(ge^{\hat{e_{i}}}\right)\right]\] | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
| which is the basis for a numerical derivative scheme. | ||||
| \end_layout | ||||
| 
 | ||||
| \begin_layout Standard | ||||
| Let us also look at a chain rule. | ||||
|  If we know the behavior at the origin  | ||||
| \begin_inset Formula $I$ | ||||
| \end_inset | ||||
| 
 | ||||
| , we can extrapolate | ||||
| \begin_inset Formula \[ | ||||
| f(ge^{\xhat})=f(ge^{\xhat}g^{-1}g)=f(e^{\Ad g\xhat}g)\] | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
|   | ||||
| \end_layout | ||||
| 
 | ||||
| \begin_layout Section | ||||
| Derivatives of Actions | ||||
| \begin_inset CommandInset label | ||||
|  | @ -594,15 +899,7 @@ Derivatives of Mappings | |||
| \end_layout | ||||
| 
 | ||||
| \begin_layout Standard | ||||
| The adjoint map for  | ||||
| \begin_inset Formula $\sotwo$ | ||||
| \end_inset | ||||
| 
 | ||||
|  is trivially equal to the identity, as is the case for  | ||||
| \emph on | ||||
| all | ||||
| \emph default | ||||
|  commutative groups, and we have the derivative of  | ||||
| We have the derivative of  | ||||
| \series bold | ||||
| inverse | ||||
| \series default | ||||
|  | @ -1065,10 +1362,10 @@ Hence, as with | |||
| inverse | ||||
| \series default | ||||
| , | ||||
| \begin_inset Formula \begin{eqnarray*} | ||||
| \frac{\partial T^{-1}}{\partial\xi} & = & -\left[\begin{array}{cc} | ||||
| \begin_inset Formula \[ | ||||
| \frac{\partial T^{-1}}{\partial\xi}=\Ad T=-\left[\begin{array}{cc} | ||||
| R & 0\\ | ||||
| \Skew tR & R\end{array}\right]\end{eqnarray*} | ||||
| \Skew tR & R\end{array}\right]\] | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
|  | @ -1077,16 +1374,18 @@ R & 0\\ | |||
| compose | ||||
| \series default | ||||
|  in its first argument, | ||||
| \begin_inset Formula \begin{eqnarray*} | ||||
| \frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{1}} & = & \left[\begin{array}{cc} | ||||
| \begin_inset Formula \[ | ||||
| \frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{1}}=\Ad{T_{2}^{-1}}=\left[\begin{array}{cc} | ||||
| R_{2}^{T} & 0\\ | ||||
| \Skew{-R_{2}^{T}t}R_{2}^{T} & R_{2}^{T}\end{array}\right]\end{eqnarray*} | ||||
| \Skew{-R_{2}^{T}t_{2}}R_{2}^{T} & R_{2}^{T}\end{array}\right]=\left[\begin{array}{cc} | ||||
| R_{2}^{T} & 0\\ | ||||
| R_{2}^{T}\Skew{-t_{2}} & R_{2}^{T}\end{array}\right]\] | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
| compose in its second argument, | ||||
| \begin_inset Formula \begin{eqnarray*} | ||||
| \frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{2}} & = & I_{6}\end{eqnarray*} | ||||
| \begin_inset Formula \[ | ||||
| \frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{2}}=I_{6}\] | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
|  | @ -1095,16 +1394,16 @@ compose in its second argument, | |||
| between | ||||
| \series default | ||||
|  in its first argument, | ||||
| \begin_inset Formula \begin{eqnarray*} | ||||
| \frac{\partial\left(T_{1}^{^{-1}}T_{2}\right)}{\partial\xi_{1}} & = & -\left[\begin{array}{cc} | ||||
| \begin_inset Formula \[ | ||||
| \frac{\partial\left(T_{1}^{^{-1}}T_{2}\right)}{\partial\xi_{1}}=\Ad{T_{21}}=-\left[\begin{array}{cc} | ||||
| R & 0\\ | ||||
| \Skew tR & R\end{array}\right]\end{eqnarray*} | ||||
| \Skew tR & R\end{array}\right]\] | ||||
| 
 | ||||
| \end_inset | ||||
| 
 | ||||
| with  | ||||
| \begin_inset Formula \[ | ||||
| \left[\begin{array}{cc} | ||||
| T_{12}=\left[\begin{array}{cc} | ||||
| R & t\\ | ||||
| 0 & 1\end{array}\right]=T_{1}^{^{-1}}T_{2}=between(T_{2},T_{1})\] | ||||
| 
 | ||||
|  |  | |||
							
								
								
									
										
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