From 8f4c1278b945851a557361d5fbc39093dedb627a Mon Sep 17 00:00:00 2001 From: Stephen Williams Date: Thu, 16 May 2013 15:24:37 +0000 Subject: [PATCH] Updated BetweenFactor section of 'math' document --- doc/math.lyx | 205 ++++++++++++++++++++++++++++++++++++++++++++------- doc/math.pdf | Bin 127548 -> 173687 bytes 2 files changed, 178 insertions(+), 27 deletions(-) diff --git a/doc/math.lyx b/doc/math.lyx index 05eb355ad..8d56963cb 100644 --- a/doc/math.lyx +++ b/doc/math.lyx @@ -885,7 +885,39 @@ H_{a}=G_{f(a)}F_{a} \end_inset +where +\begin_inset Formula $G_{f(a)}$ +\end_inset + is the +\begin_inset Formula $m\times p$ +\end_inset + + Jacobian matrix of +\begin_inset Formula $g$ +\end_inset + + evaluated at +\begin_inset Formula $f(a)$ +\end_inset + +, and +\begin_inset Formula $F_{a}$ +\end_inset + + is the +\begin_inset Formula $p\times n$ +\end_inset + + Jacobian matrix of +\begin_inset Formula $f$ +\end_inset + + evaluated at +\begin_inset Formula $a$ +\end_inset + +. \end_layout \begin_layout Proof @@ -1521,7 +1553,7 @@ name "th:Action" \end_inset The Jacobian matrix of the group action -\begin_inset Formula $f(T,P)=Tp$ +\begin_inset Formula $f(T,p)=Tp$ \end_inset at @@ -2811,6 +2843,12 @@ B^{T} & I_{3}\end{array}\right] \end_layout \begin_layout Subsection +\begin_inset CommandInset label +LatexCommand label +name "sub:Pushforward-of-Between" + +\end_inset + Pushforward of Between \end_layout @@ -3378,27 +3416,14 @@ BetweenFactor \end_layout \begin_layout Standard -BetweenFactor is often used to summarize -\end_layout -\begin_layout Standard -Theorem -\begin_inset CommandInset ref -LatexCommand ref -reference "D-exp" - -\end_inset - - about the derivative of the exponential map -\begin_inset Formula $f:\xi\mapsto\exp\xihat$ -\end_inset - - being identity only at -\begin_inset Formula $\xi=0$ -\end_inset - - has implications for GTSAM. - Given two elements +\series bold +\emph on +BetweenFactor +\series default +\emph default + is a factor in GTSAM that is used ubiquitously to process measurements + indicating the relative pose between two unknown poses \begin_inset Formula $T_{1}$ \end_inset @@ -3406,15 +3431,104 @@ reference "D-exp" \begin_inset Formula $T_{2}$ \end_inset -, BetweenFactor evaluates +. + Let us assume the measured relative pose is +\begin_inset Formula $Z$ +\end_inset + +, then the code that calculates the error in +\series bold +\emph on +BetweenFactor +\series default +\emph default + first calculates the predicted relative pose +\begin_inset Formula $T_{12}$ +\end_inset + +, and then evaluates the error between the measured and predicted relative + pose: +\end_layout + +\begin_layout LyX-Code +T12 = between(T1, T2); +\end_layout + +\begin_layout LyX-Code +return localCoordinates(Z, T12); +\end_layout + +\begin_layout Standard +where we recall that the function +\series bold +\emph on +between +\series default +\emph default + is given in group theoretic notation as \begin_inset Formula \[ -g(T_{1},T_{2};Z)=f^{-1}\left(\mathop{between}(Z,\mathop{between}(T_{1},T_{2})\right)=f^{-1}\left(Z^{-1}\left(T_{1}^{-1}T_{2}\right)\right) +\varphi(g,h)=g^{-1}h \] \end_inset -but because it is assumed that +The function +\series bold +\emph on +localCoordinates +\series default +\emph default + itself also calls +\series bold +\emph on +between +\series default +\emph default +, and converts to canonical coordinates: +\end_layout + +\begin_layout LyX-Code +localCoordinates(Z,T12) = Logmap(between(Z, T12)); +\end_layout + +\begin_layout Standard +Hence, given two elements +\begin_inset Formula $T_{1}$ +\end_inset + + and +\begin_inset Formula $T_{2}$ +\end_inset + +, +\series bold +\emph on +BetweenFactor +\series default +\emph default + evaluates +\begin_inset Formula $g:G\times G\rightarrow\Reals n$ +\end_inset + +, +\begin_inset Formula +\[ +g(T_{1},T_{2};Z)=f^{-1}\left(\varphi(Z,\varphi(T_{1},T_{2})\right)=f^{-1}\left(Z^{-1}\left(T_{1}^{-1}T_{2}\right)\right) +\] + +\end_inset + +where +\begin_inset Formula $f^{-1}$ +\end_inset + + is the inverse of the map +\begin_inset Formula $f:\xi\mapsto\exp\xihat$ +\end_inset + +. + If we assume that the measurement has only small error, then \begin_inset Formula $Z\approx T_{1}^{-1}T_{2}$ \end_inset @@ -3422,12 +3536,49 @@ but because it is assumed that \begin_inset Formula $Z^{-1}T_{1}^{-1}T_{2}\approx e$ \end_inset - and the derivative should be good there. - Note that the derivative of +, and we can invoke Theorem +\begin_inset CommandInset ref +LatexCommand ref +reference "D-exp" + +\end_inset + +, which says that the derivative of the exponential map +\begin_inset Formula $f:\xi\mapsto\exp\xihat$ +\end_inset + + is identity at +\begin_inset Formula $\xi=0$ +\end_inset + +, as well as its inverse. +\end_layout + +\begin_layout Standard +Finally, because the derivative of +\series bold \emph on between +\series default \emph default - is identity in its second argument. + is identity in its second argument, the derivative of the +\series bold +\emph on +BetweenFactor +\series default +\emph default + error is identical to the derivative of pushforward of +\begin_inset Formula $\varphi(T_{1},T_{2})$ +\end_inset + +, derived in Section +\begin_inset CommandInset ref +LatexCommand ref +reference "sub:Pushforward-of-Between" + +\end_inset + +. \end_layout \begin_layout Section diff --git a/doc/math.pdf b/doc/math.pdf index eb429095ca9e3467baa95e577b53fde0516defea..9b1046f68c4dccaa64b02e2792dbae722fc233a2 100644 GIT binary patch literal 173687 zcma&sQ*$mryQu5fwr$(CZQHi(5!-e~Y}>YzH@2M-tZ&X*v-ZI}*z2Ub>-hm))mPu- zN@5c9%nYnBw1A3kblty1AGe+rxO}?&?XW;7cNVWWqB42I;!`B0gttVM;nNb6`b6hw>xi*&!^= zNSU_GSYiKN+z%p@vryMm*Kl`#^Yu~&D+<;5wwavb@1I(m4vzCPdbj-iclfb+*Vi>QUB*`c>IiM06rvc42m^ZP~Z>h$3Px_8CL9TiPP}MQ?sK-!S%;^ zi?fMmD`pAuPUkMI*I8Uz_uZ>V>=ak?zjcF&MILJp#HR+%f1tRvixN-PrELTfF0jp6 zl(2n!gkftXN$GNdjg}=ETiku!Bm1l8!7 zTu3f}5|1G(Ednn``nq3`eID9_8&$qCEPPHUnGHj2$Dm8TXYBkD7A)-i*e7f9`OFg1 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