update Lyx document based on Luca's review
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@ -281,8 +281,8 @@ The noise model associated with this factor is assumed to be zero-mean Gaussian
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\end_inset
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.
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This covariance matrix is computed in the preintegrated measurement class,
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of which there are two variants as discussed above.
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This (discrete-time) covariance matrix is computed in the preintegrated
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measurement class, of which there are two variants as discussed above.
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\end_layout
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\begin_layout Subsubsection*
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@ -309,6 +309,20 @@ CombinedImuFactor2 is a 4-way factor between the previous NavState and IMU
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bias and the current NavState and IMU bias.
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\end_layout
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\begin_layout Standard
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Since the Combined IMU Factor has a larger state variable due to the inclusion
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of IMU biases, the noise model associated with this factor is assumed to
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be a zero mean Gaussian with a
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\begin_inset Formula $15\times15$
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\end_inset
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covariance matrix
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\begin_inset Formula $\Sigma$
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\end_inset
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, similarly defined on the tangent space of the NavState manifold.
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\end_layout
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\begin_layout Subsubsection*
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Covariance Matrices
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\end_layout
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@ -564,7 +578,15 @@ acceleration
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\end_inset
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in the body frame.
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We know (from Murray84book) that the derivative of
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We know (from
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\begin_inset CommandInset citation
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LatexCommand cite
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key "Murray94book"
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literal "false"
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\end_inset
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) that the derivative of
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\begin_inset Formula $R$
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\end_inset
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@ -1618,6 +1640,42 @@ where
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\begin_inset Formula $\omega^{b}$
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\end_inset
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.
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Note that
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\begin_inset Formula $\Sigma_{k},$
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\end_inset
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\begin_inset Formula $\Sigma_{\eta}^{ad}$
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\end_inset
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, and
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\begin_inset Formula $\Sigma_{\eta}^{gd}$
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\end_inset
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are discrete time covariances with
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\begin_inset Formula $\Sigma_{\eta}^{ad}$
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\end_inset
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, and
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\begin_inset Formula $\Sigma_{\eta}^{gd}$
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\end_inset
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divided by
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\begin_inset Formula $\Delta_{t}$
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\end_inset
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.
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Please see the section on Covariance Discretization
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\begin_inset CommandInset ref
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LatexCommand vpageref
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reference "subsec:Covariance-Discretization"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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.
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\end_layout
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@ -1645,7 +1703,7 @@ It can be shown that for small
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we have
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\begin_inset Formula
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\[
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\deriv{H(\theta_{k})^{-1}\omega_{k}^{b}}{\theta_{k}}\approx-\frac{1}{2}\Skew{\omega_{k}^{b}}\mbox{ and hence }\deriv{\theta_{k+1}}{\theta_{k}}=I_{3\times3}-\frac{\Delta t}{2}\Skew{\omega_{k}^{b}}
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\deriv{H(\theta_{k})^{-1}\omega_{k}^{b}}{\theta_{k}}\approx-\frac{1}{2}\Skew{\omega_{k}^{b}}\mbox{ and hence }\deriv{\theta_{k+1}}{\theta_{k}}=I_{3\times3}-\frac{\Delta_{t}}{2}\Skew{\omega_{k}^{b}}
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\]
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\end_inset
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@ -2033,6 +2091,13 @@ which we can break into 3 matrices for clarity, representing the main diagonal
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\begin_layout Subsubsection*
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Covariance Discretization
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\begin_inset CommandInset label
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LatexCommand label
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name "subsec:Covariance-Discretization"
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\end_inset
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\end_layout
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\begin_layout Standard
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