Math doc now matches code but doc is fairly rough right now
Richard Roberts
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doc/math.lyx
244
doc/math.lyx
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@ -64,6 +64,106 @@ This document should be kept up to date and specify how each of the derivatives
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in the geometry modules are computed.
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\end_layout
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\begin_layout Section
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General Lie group derivations
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\end_layout
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\begin_layout Standard
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The derivatives for
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\emph on
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compose
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\emph default
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,
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\emph on
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inverse
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\emph default
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, and
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\emph on
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between
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\emph default
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can be derived from Lie group principals to work with any transformation
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type.
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To find the derivatives of these functions, we look for the necessary
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\begin_inset Quotes eld
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\end_inset
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delta
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\begin_inset Quotes erd
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\end_inset
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in the tangent space of the function
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\emph on
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output
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\emph default
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that corresponds to a
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\begin_inset Quotes eld
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\end_inset
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delta
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\begin_inset Quotes erd
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\end_inset
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in the tangent space of the function
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\emph on
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input
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\emph default
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.
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For example, to find the derivative of a function
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\begin_inset Formula $f\left(X\right)$
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\end_inset
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, we include the differential changes in the tangent space:
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\begin_inset Formula \[
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f\left(X\right)\exp\partial f=f\left(X\exp\partial x\right)\]
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\end_inset
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and then taking the partial derivatives
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\begin_inset Formula $\frac{\partial y}{\partial x}$
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\end_inset
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.
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Calculating these derivatives requires that we know the form of the function
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\begin_inset Formula $f$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\series bold
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\emph on
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This section is not correct - math doesn't make sense and need to fix.
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\end_layout
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\begin_layout Standard
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Starting with inverse:
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\begin_inset Formula \begin{align*}
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X^{-1}\exp\partial i & =\left(X\exp\left[\partial x\right]\right)^{-1}\\
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& =\left(\exp-\left[\partial x\right]\right)X^{-1}\\
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\exp\partial i & =X\left(\exp-\left[\partial x\right]\right)X^{-1}\\
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& =\exp-X\left[\partial x\right]X^{-1}\\
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\partial i & =-X\left[\partial x\right]X^{-1}\end{align*}
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\end_inset
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\end_layout
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\begin_layout Standard
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Compose can be derived similarly:
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\begin_inset Formula \begin{align*}
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AB\exp\partial c & =A\left(\exp\left[\partial a\right]\right)B\\
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\exp\partial c & =B^{-1}\left(\exp\left[\partial a\right]\right)B\\
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\partial c & =B^{-1}\left[\partial a\right]B\end{align*}
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\end_inset
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\end_layout
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\begin_layout Section
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Rot2 (in gtsam)
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\end_layout
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@ -453,7 +553,7 @@ In the old-style, we have
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\bar no
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\noun off
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\color none
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\begin_inset Formula $R'=(I+\Omega)R$
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\begin_inset Formula $R'=R(I+\Omega)$
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\end_inset
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@ -467,7 +567,13 @@ In the old-style, we have
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\end_layout
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\begin_layout Standard
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In this case, the derivative of transform_from,
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In this case, the derivative of
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\series bold
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\emph on
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transform_from
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\series default
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\emph default
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,
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\begin_inset Formula $Rx+t$
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\end_inset
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@ -476,7 +582,7 @@ In this case, the derivative of transform_from,
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\begin_layout Standard
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\begin_inset Formula \[
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\frac{\partial((I+\Omega)Rx+t)}{\partial\omega}=\frac{\partial(\Omega Rx)}{\partial\omega}=\frac{\partial(\omega\times Rx)}{\partial\omega}=-\Skew{Rx}\]
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\frac{\partial(R(I+\Omega)x+t)}{\partial\omega}=\frac{\partial(R\Omega x)}{\partial\omega}=\frac{\partial(R\left(\omega\times x\right))}{\partial\omega}=R\Skew{-x}\]
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\end_inset
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@ -493,32 +599,26 @@ and with respect to
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\end_inset
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The derivative of transform_to,
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\begin_inset Formula $R^{T}(x-t)$
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The derivative of
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\series bold
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\emph on
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transform_to
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\series default
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\emph default
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,
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\begin_inset Formula $inv(R)(x-t)$
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\end_inset
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, noting that
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\family roman
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\series medium
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\shape up
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\size normal
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\emph off
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\bar no
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\noun off
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\color none
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we can obtain using the chain rule:
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\begin_inset Formula \[
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\frac{\partial(inv(R)(x-t))}{\partial\omega}=\frac{\partial unrot(R,(x-t))}{\partial\omega}=skew(R^{T}\left(x-t\right))\]
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\begin_inset Formula $R'^{T}=R^{T}(I-\Omega)$
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\end_inset
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, is
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\end_layout
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\begin_layout Standard
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\begin_inset Formula \[
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\frac{\partial(R'^{T}(x-t))}{\partial\omega}=\frac{\partial(R^{T}(I-\Omega)(x-t))}{\partial\omega}=-\frac{\partial(R^{T}\Omega(x-t))}{\partial\omega}=-\frac{\partial(\Skew{R^{T}\omega}R^{T}(x-t))}{\partial\omega}=\Skew{R^{T}(x-t)}\frac{\partial(R^{T}\omega)}{\partial\omega}=\Skew{R^{T}(x-t)}R^{T}\]
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\end_inset
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and with respect to
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\begin_inset Formula $dt$
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\end_inset
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@ -541,6 +641,38 @@ The derivative of
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\end_inset
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, first derivative of rotation in rotation argument:
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\end_layout
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\begin_layout Standard
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The partials
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\begin_inset Formula \[
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\frac{\partial\omega^{\prime}}{\partial\omega}=\frac{\partial inv(R)}{\partial\omega}=-R\]
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\end_inset
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\begin_inset Formula \[
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\frac{\partial t^{\prime}}{\partial\omega}=\frac{-\partial unrot(R,t)}{\partial\omega}=-skew(R^{T}t)\]
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\end_inset
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\begin_inset Formula \[
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\frac{\partial\omega^{\prime}}{\partial t}=\mathbf{0}\]
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\end_inset
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\begin_inset Formula \[
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\frac{\partial t^{\prime}}{\partial t}=\frac{-\partial unrot(R,t)}{\partial t}=-R^{T}\]
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\end_inset
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\series bold
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old stuff:
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\series default
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\begin_inset Formula \begin{eqnarray*}
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(I+\Omega')R^{T} & = & \left((I+\Omega)R\right)^{T}\\
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R^{T}+\Omega'R^{T} & = & R^{T}(I-\Omega)\\
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@ -551,6 +683,76 @@ R^{T}+\Omega'R^{T} & = & R^{T}(I-\Omega)\\
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\end_inset
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\end_layout
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\begin_layout Standard
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Now
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\series bold
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\emph on
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compose
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\series default
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\emph default
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, first w.r.t.
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a change in rotation in the first argument:
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\begin_inset Formula \begin{align*}
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AB & =\left(T_{A}R_{A}T_{B}\right)\left(R_{A}R_{B}\right)\\
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\left(T_{A}R_{A}T_{B}\left(I+T^{\prime}\right)\right)\left(R_{A}R_{B}\left(I+\Omega^{\prime}\right)\right) & =\left(T_{A}R_{A}\left(I+\Omega\right)T_{B}\right)\left(R_{A}\left(I+\Omega\right)R_{B}\right)\\
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\textrm{translation only:}\\
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T_{A}R_{A}T_{B}\left(I+T^{\prime}\right) & =T_{A}R_{A}\left(I+\Omega\right)T_{B}\\
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T_{B}\left(I+T^{\prime}\right) & =\left(I+\Omega\right)T_{B}\\
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T_{B}+T_{B}T^{\prime} & =T_{B}+\Omega T_{B}\\
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T^{\prime} & =T_{B}^{-1}skew(\omega)T_{B}\\
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T^{\prime} & =skew(T_{B}\omega)\,???\\
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\textrm{rotation only:}\\
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R_{A}R_{B}\left(I+\Omega^{\prime}\right) & =R_{A}\left(I+\Omega\right)R_{B}\\
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R_{B}\Omega^{\prime} & =\Omega R_{B}\\
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\Omega^{\prime} & =R_{B}^{T}\Omega R_{B}\\
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& =skew(R_{B}^{T}\omega)\\
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\omega^{\prime} & =R_{B}^{T}\omega\end{align*}
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\end_inset
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\end_layout
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\begin_layout Standard
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And w.r.t.
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a rotation in the second argument:
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\begin_inset Formula \begin{align*}
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\left(T_{A}R_{A}T_{B}\left(I+T^{\prime}\right)\right)\left(R_{A}R_{B}\left(I+\Omega^{\prime}\right)\right) & =\left(T_{A}R_{A}T_{B}\right)\left(R_{A}R_{B}\left(I+\Omega\right)\right)\\
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\left(R_{A}R_{B}\left(I+\Omega^{\prime}\right)\right) & =\left(R_{A}R_{B}\left(I+\Omega\right)\right)\\
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\omega^{\prime} & =\omega\\
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t^{\prime} & =0\end{align*}
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\end_inset
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w.r.t.
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a translation in the second argument:
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\begin_inset Formula \begin{align*}
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\left(T_{A}R_{A}T_{B}\left(I+T^{\prime}\right)\right)\left(R_{A}R_{B}\left(I+\Omega^{\prime}\right)\right) & =\left(T_{A}R_{A}T_{B}\left(I+T\right)\right)\left(R_{A}R_{B}\right)\\
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\omega^{\prime} & =0\\
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t^{\prime} & =t\end{align*}
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\end_inset
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\end_layout
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\begin_layout Standard
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Finally,
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\series bold
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\emph on
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between
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\series default
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\emph default
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in the first argument:
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\begin_inset Formula \begin{align*}
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\frac{\partial A^{-1}B}{\partial A} & =\frac{\partial c\left(A^{-1},B\right)}{\partial A^{-1}}\frac{\partial inv(A)}{A}\\
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\frac{\partial A^{-1}B}{B} & =\frac{\partial c\left(A^{-1},B\right)}{\partial B}\end{align*}
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\end_inset
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\end_layout
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\begin_layout Section
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