add details about noise propagation for CombinedImuFactor in ImuFactor.pdf

doc/ImuFactor.lyx

@@ -1,7 +1,9 @@
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@@ -29,16 +33,26 @@
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@@ -51,7 +65,10 @@
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@@ -69,7 +86,7 @@ The new IMU Factor
 \end_layout
 
 \begin_layout Author
-Frank Dellaert
+Frank Dellaert & Varun Agrawal
 \end_layout
 
 \begin_layout Standard
@@ -592,6 +609,7 @@ Lie Group Methods
 \begin_inset CommandInset citation
 LatexCommand cite
 key "Iserles00an"
+literal "true"
 
 \end_inset
 
@@ -1382,6 +1400,99 @@ H(\theta_{k})^{-1}\Delta_{t}\\
 \end_inset
 
 
+\end_layout
+
+\begin_layout Subsubsection*
+Combined IMU Factor
+\end_layout
+
+\begin_layout Standard
+We can similarly account for bias drift over time, as is commonly seen in
+ commercial grade IMUs.
+ This is accomplished via the 
+\emph on
+CombinedImuFactor
+\emph default
+ which is a 6-way factor between the previous 
+\emph on
+pose/velocity/bias
+\emph default
+ and the 
+\emph on
+pose/velocity/bias
+\emph default
+ at the next timestep.
+ 
+\end_layout
+
+\begin_layout Standard
+We expand the state vector as 
+\begin_inset Formula $\zeta_{k}=[\theta_{k},p_{k},v_{k},a_{k}^{b}, \omega_{k}^{b}]$
+\end_inset
+
+.
+ For the jacobian 
+\begin_inset Formula $F_{k}$
+\end_inset
+
+ of 
+\begin_inset Formula $f$
+\end_inset
+
+ wrpt this new 
+\begin_inset Formula $\zeta$
+\end_inset
+
+, we get a 
+\begin_inset Formula $15\times15$
+\end_inset
+
+ matrix.
+ The top-left 
+\begin_inset Formula $9\times9$
+\end_inset
+
+ is the same as 
+\begin_inset Formula $A_{k}$
+\end_inset
+
+, thus we only have the jacobians wrpt the biases left to account for.
+\end_layout
+
+\begin_layout Standard
+Conveniently, the jacobians of the pose and velocity wrpt the biases are
+ already computed in the 
+\emph on
+ImuFactor 
+\emph default
+derivation as matrices 
+\begin_inset Formula $B_{k}$
+\end_inset
+
+ and 
+\begin_inset Formula $C_{k}$
+\end_inset
+
+, while they are identity matrices wrpt the biases themselves.
+ Thus, we can easily plug-in the values from the previous section to give
+ us the final result
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+F_{k}=\left[\begin{array}{ccccc}
+I_{3\times3}-\frac{\Delta_{t}}{2}\Skew{\omega_{k}^{b}} &  &  &  & H(\theta_{k})^{-1}\Delta_{t}\\
+R_{k}\Skew{-a_{k}^{b}}H(\theta_{k})\frac{\Delta_{t}}{2}^{2} & I_{3\times3} & I_{3\times3}\Delta_{t} & R_{k}\frac{\Delta_{t}}{2}^{2}\\
+R_{k}\Skew{-a_{k}^{b}}H(\theta_{k})\Delta_{t} &  & I_{3\times3} & R_{k}\Delta_{t}\\
+ &  &  & I_{3\times3}\\
+ &  &  &  & I_{3\times3}
+\end{array}\right]
+\]
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard