From bbd1e3f07a9e8b03322d9a1b51efded5f123956f Mon Sep 17 00:00:00 2001 From: Varun Agrawal Date: Sun, 3 Jul 2022 15:57:54 -0400 Subject: [PATCH] update Lyx document based on Luca's review --- doc/ImuFactor.lyx | 73 +++++++++++++++++++++++++++++++++++++++++++--- doc/ImuFactor.pdf | Bin 249424 -> 250191 bytes 2 files changed, 69 insertions(+), 4 deletions(-) diff --git a/doc/ImuFactor.lyx b/doc/ImuFactor.lyx index 7b176f484e..c335e69496 100644 --- a/doc/ImuFactor.lyx +++ b/doc/ImuFactor.lyx @@ -281,8 +281,8 @@ The noise model associated with this factor is assumed to be zero-mean Gaussian \end_inset . - This covariance matrix is computed in the preintegrated measurement class, - of which there are two variants as discussed above. + This (discrete-time) covariance matrix is computed in the preintegrated + measurement class, of which there are two variants as discussed above. \end_layout \begin_layout Subsubsection* @@ -309,6 +309,20 @@ CombinedImuFactor2 is a 4-way factor between the previous NavState and IMU bias and the current NavState and IMU bias. \end_layout +\begin_layout Standard +Since the Combined IMU Factor has a larger state variable due to the inclusion + of IMU biases, the noise model associated with this factor is assumed to + be a zero mean Gaussian with a +\begin_inset Formula $15\times15$ +\end_inset + + covariance matrix +\begin_inset Formula $\Sigma$ +\end_inset + +, similarly defined on the tangent space of the NavState manifold. +\end_layout + \begin_layout Subsubsection* Covariance Matrices \end_layout @@ -564,7 +578,15 @@ acceleration \end_inset in the body frame. - We know (from Murray84book) that the derivative of + We know (from +\begin_inset CommandInset citation +LatexCommand cite +key "Murray94book" +literal "false" + +\end_inset + +) that the derivative of \begin_inset Formula $R$ \end_inset @@ -1618,6 +1640,42 @@ where \begin_inset Formula $\omega^{b}$ \end_inset +. + Note that +\begin_inset Formula $\Sigma_{k},$ +\end_inset + + +\begin_inset Formula $\Sigma_{\eta}^{ad}$ +\end_inset + +, and +\begin_inset Formula $\Sigma_{\eta}^{gd}$ +\end_inset + + are discrete time covariances with +\begin_inset Formula $\Sigma_{\eta}^{ad}$ +\end_inset + +, and +\begin_inset Formula $\Sigma_{\eta}^{gd}$ +\end_inset + +divided by +\begin_inset Formula $\Delta_{t}$ +\end_inset + +. + Please see the section on Covariance Discretization +\begin_inset CommandInset ref +LatexCommand vpageref +reference "subsec:Covariance-Discretization" +plural "false" +caps "false" +noprefix "false" + +\end_inset + . \end_layout @@ -1645,7 +1703,7 @@ It can be shown that for small we have \begin_inset Formula \[ -\deriv{H(\theta_{k})^{-1}\omega_{k}^{b}}{\theta_{k}}\approx-\frac{1}{2}\Skew{\omega_{k}^{b}}\mbox{ and hence }\deriv{\theta_{k+1}}{\theta_{k}}=I_{3\times3}-\frac{\Delta t}{2}\Skew{\omega_{k}^{b}} +\deriv{H(\theta_{k})^{-1}\omega_{k}^{b}}{\theta_{k}}\approx-\frac{1}{2}\Skew{\omega_{k}^{b}}\mbox{ and hence }\deriv{\theta_{k+1}}{\theta_{k}}=I_{3\times3}-\frac{\Delta_{t}}{2}\Skew{\omega_{k}^{b}} \] \end_inset @@ -2033,6 +2091,13 @@ which we can break into 3 matrices for clarity, representing the main diagonal \begin_layout Subsubsection* Covariance Discretization +\begin_inset CommandInset label +LatexCommand label +name "subsec:Covariance-Discretization" + +\end_inset + + \end_layout \begin_layout Standard diff --git a/doc/ImuFactor.pdf b/doc/ImuFactor.pdf index 9d7026d0978944949f5f6d4932c85dbe112a2a10..1823cbc4a6ea54b1b6a5aa5dae7b7909ba75a784 100644 GIT binary patch delta 37371 zcmV(_K-9m`oDa{U50E4RG$50)V<>+ukIiZWF${$7`xH7A8@j3`{~uG@g#_A@+e^s7 zw3`&Nq+k+4-@bO7i!=k$r_pm7o+kD{9+|9xvJFC>+H3aytGSTH7e&|WK(0hj^13BzrLX#JDn9v5`53*fD}d*>Go(q z)3RB^=-6}uwaMW%4gZLQL6?&j0Ve}DH8_*8V<>;U8C$R0w)wrk!jCo(t0F0qQii=` zScVkHP^@RbfVLPETPH`UZF!QMH2d}YZj>nq@+X+ysLu-U>yGvYKYY?>3# zj7R_u&VBQ_?Jk4-!G?0(4Ki7u_B{AZn-6~-P4C(ff4?GjuVCIp$L%hC-!Pes=$TN! z6NTmyc({eHwM;YJ!CXRI=?>3#_~{OumxY=yL&)qhoDpfL^@p||GP-dauGA_sh2uN* 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