diff --git a/doc/math.lyx b/doc/math.lyx index 6d7a5e318..4ee89a9cc 100644 --- a/doc/math.lyx +++ b/doc/math.lyx @@ -5086,6 +5086,13 @@ reference "ex:projection" \begin_layout Subsection Derivative of Adjoint +\begin_inset CommandInset label +LatexCommand label +name "subsec:pose3_adjoint_deriv" + +\end_inset + + \end_layout \begin_layout Standard @@ -5098,7 +5105,7 @@ Consider \end_inset . - The derivative is notated (see + The derivative is notated (see Section \begin_inset CommandInset ref LatexCommand ref reference "sec:Derivatives-of-Actions" @@ -5114,7 +5121,7 @@ noprefix "false" \begin_layout Standard \begin_inset Formula \[ -Df_{(T,y)}(\xi,\delta y)=D_{1}f_{(T,y)}(\xi)+D_{2}f_{(T,y)}(\delta y) +Df_{(T,\xi_{b})}(\xi,\delta\xi_{b})=D_{1}f_{(T,\xi_{b})}(\xi)+D_{2}f_{(T,\xi_{b})}(\delta\xi_{b}) \] \end_inset @@ -5149,11 +5156,12 @@ D_{2}f_{(T,\xi_{b})}(\xi_{b})=Ad_{T} \end_inset -To compute -\begin_inset Formula $D_{1}f_{(T,\xi_{b})}(\xi_{b})$ +We will derive +\begin_inset Formula $D_{1}f_{(T,\xi_{b})}(\xi)$ \end_inset -, we'll first define + using two approaches. + In the first, we'll define \begin_inset Formula $g(T,\xi)\triangleq T\exp\hat{\xi}$ \end_inset @@ -5194,18 +5202,30 @@ Now we can use the definition of the Adjoint representation \begin_layout Standard \begin_inset Formula \begin{align*} -D_{1}f_{(T,\xi_{b})}(\xi)=D_{1}\left(Ad_{T\exp(\hat{\xi})}\hat{\xi}_{b}\right)(\xi) & =D_{1}\left(g\hat{\xi}g^{-1}\right)(\xi)\\ - & =\left(D_{2}g_{(T,\xi)}(\xi)\right)\hat{\xi}g^{-1}(T,0)+g(T,0)\hat{\xi}\left(D_{2}g_{(T,\xi)}^{-1}(\xi)\right)\\ +D_{1}f_{(T,\xi_{b})}(\xi)=D_{1}\left(Ad_{T\exp(\hat{\xi})}\hat{\xi}_{b}\right)(\xi) & =D_{1}\left(g\hat{\xi}_{b}g^{-1}\right)(\xi)\\ + & =\left(D_{2}g_{(T,\xi)}(\xi)\right)\hat{\xi}_{b}g^{-1}(T,0)+g(T,0)\hat{\xi}_{b}\left(D_{2}g_{(T,\xi)}^{-1}(\xi)\right)\\ & =T\hat{\xi}\hat{\xi}_{b}T^{-1}-T\hat{\xi}_{b}\hat{\xi}T^{-1}\\ & =T\left(\hat{\xi}\hat{\xi}_{b}-\hat{\xi}_{b}\hat{\xi}\right)T^{-1}\\ - & =-Ad_{T}(ad_{\xi_{b}}\hat{\xi})\\ + & =Ad_{T}(ad_{\hat{\xi}}\hat{\xi}_{b})\\ + & =-Ad_{T}(ad_{\hat{\xi}_{b}}\hat{\xi})\\ D_{1}F_{(T,\xi_{b})} & =-(Ad_{T})(ad_{\hat{\xi}_{b}}) \end{align*} \end_inset -An alternative, perhaps more intuitive way of deriving this would be to - use the fact that the derivative at the origin +Where +\begin_inset Formula $ad_{\hat{\xi}}:\mathfrak{g}\rightarrow\mathfrak{g}$ +\end_inset + + is the adjoint map of the lie algebra. +\end_layout + +\begin_layout Standard +The second, perhaps more intuitive way of deriving +\begin_inset Formula $D_{1}f_{(T,\xi_{b})}(\xi_{b})$ +\end_inset + +, would be to use the fact that the derivative at the origin \begin_inset Formula $D_{1}Ad_{I}\hat{\xi}_{b}=ad_{\hat{\xi}_{b}}$ \end_inset @@ -5224,28 +5244,212 @@ An alternative, perhaps more intuitive way of deriving this would be to \begin_layout Standard \begin_inset Formula \[ -D_{1}Ad_{T}\hat{\xi}_{b}(\xi)=D_{1}Ad_{T*I}\hat{\xi}_{b}(\xi)=Ad_{T}\left(D_{1}Ad_{I}\hat{\xi}_{b}(\xi)\right)=Ad_{T}\left(ad_{\hat{\xi}_{b}}(\xi)\right) +D_{1}Ad_{T}\hat{\xi}_{b}(\xi)=D_{1}Ad_{T*I}\hat{\xi}_{b}(\xi)=Ad_{T}\left(D_{1}Ad_{I}\hat{\xi}_{b}(\xi)\right)=Ad_{T}\left(ad_{\hat{\xi}}(\hat{\xi}_{b})\right)=-Ad_{T}\left(ad_{\hat{\xi}_{b}}(\hat{\xi})\right) \] \end_inset -It's difficult to apply a similar procedure to compute the derivative of - -\begin_inset Formula $Ad_{T}^{T}\hat{\xi}_{b}^{*}$ + +\end_layout + +\begin_layout Subsection +Derivative of AdjointTranspose +\end_layout + +\begin_layout Standard +The transpose of the Adjoint, +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $Ad_{T}^{T}:\mathfrak{g^{*}\rightarrow g^{*}}$ \end_inset - (note the +, is useful as a way to change the reference frame of vectors in the dual + space +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +(note the \begin_inset Formula $^{*}$ \end_inset - denoting that we are now in the dual space) because + denoting that we are now in the dual space) +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +. + To be more concrete, where +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +as +\begin_inset Formula $Ad_{T}\hat{\xi}_{b}$ +\end_inset + + converts the +\emph on +twist +\emph default + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $\xi_{b}$ +\end_inset + + from the +\begin_inset Formula $T$ +\end_inset + + frame, +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $Ad_{T}^{T}\hat{\xi}_{b}^{*}$ +\end_inset + + converts the +\family default +\series default +\shape default +\size default +\emph on +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +wrench +\emph default + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $\xi_{b}^{*}$ +\end_inset + + from the +\begin_inset Formula $T$ +\end_inset + + frame +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +. + It's difficult to apply a similar derivation as in Section +\begin_inset CommandInset ref +LatexCommand ref +reference "subsec:pose3_adjoint_deriv" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + for the derivative of +\begin_inset Formula $Ad_{T}^{T}\hat{\xi}_{b}^{*}$ +\end_inset + + because \begin_inset Formula $Ad_{T}^{T}$ \end_inset - cannot be naturally defined as a conjugation so we resort to crunching + cannot be naturally defined as a conjugation, so we resort to crunching through the algebra. - The details are omitted but the result is a form vaguely resembling (but - not quite) the + The details are omitted but the result is a form that vaguely resembles + (but does not exactly match) \begin_inset Formula $ad(Ad_{T}^{T}\hat{\xi}_{b}^{*})$ \end_inset diff --git a/doc/math.pdf b/doc/math.pdf index 71f9dadc6..40980354e 100644 Binary files a/doc/math.pdf and b/doc/math.pdf differ