From 810f97305fa53be6d24d1945b234c2cbde652fec Mon Sep 17 00:00:00 2001 From: Varun Agrawal Date: Fri, 17 Sep 2021 14:21:39 -0400 Subject: [PATCH] add details about noise propagation for CombinedImuFactor in ImuFactor.pdf --- doc/ImuFactor.lyx | 155 +++++++++++++++++++++++++++++++++++++++------- doc/ImuFactor.pdf | Bin 198168 -> 225325 bytes 2 files changed, 133 insertions(+), 22 deletions(-) diff --git a/doc/ImuFactor.lyx b/doc/ImuFactor.lyx index 0922a3e9c..bba8f212a 100644 --- a/doc/ImuFactor.lyx +++ b/doc/ImuFactor.lyx @@ -1,7 +1,9 @@ -#LyX 2.0 created this file. For more info see http://www.lyx.org/ -\lyxformat 413 +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 \begin_document \begin_header +\save_transient_properties true +\origin unavailable \textclass article \use_default_options true \maintain_unincluded_children false @@ -9,16 +11,18 @@ \language_package default \inputencoding auto \fontencoding global -\font_roman default -\font_sans default -\font_typewriter default +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false -\font_sf_scale 100 -\font_tt_scale 100 - +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 @@ -29,16 +33,26 @@ \use_hyperref false \papersize default \use_geometry true -\use_amsmath 1 -\use_esint 1 -\use_mhchem 1 -\use_mathdots 1 +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 \cite_engine basic +\cite_engine_type default +\biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false +\justification true \use_refstyle 1 +\use_minted 0 \index Index \shortcut idx \color #008000 @@ -51,7 +65,10 @@ \tocdepth 3 \paragraph_separation indent \paragraph_indentation default -\quotes_language english +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default @@ -69,7 +86,7 @@ The new IMU Factor \end_layout \begin_layout Author -Frank Dellaert +Frank Dellaert & Varun Agrawal \end_layout \begin_layout Standard @@ -244,7 +261,7 @@ X(t)=\left\{ R_{0},P_{0}+V_{0}t,V_{0}\right\} then the differential equation describing the trajectory is \begin_inset Formula \[ -\dot{X}(t)=\left[0_{3x3},V_{0},0_{3x1}\right],\,\,\,\,\, X(0)=\left\{ R_{0},P_{0},V_{0}\right\} +\dot{X}(t)=\left[0_{3x3},V_{0},0_{3x1}\right],\,\,\,\,\,X(0)=\left\{ R_{0},P_{0},V_{0}\right\} \] \end_inset @@ -592,6 +609,7 @@ Lie Group Methods \begin_inset CommandInset citation LatexCommand cite key "Iserles00an" +literal "true" \end_inset @@ -602,7 +620,7 @@ key "Iserles00an" , \begin_inset Formula \begin{equation} -\dot{R}(t)=F(R,t),\,\,\,\, R(0)=R_{0}\label{eq:diffSo3} +\dot{R}(t)=F(R,t),\,\,\,\,R(0)=R_{0}\label{eq:diffSo3} \end{equation} \end_inset @@ -947,8 +965,8 @@ Or, as another way to state this, if we solve the differential equations \begin_inset Formula \begin{eqnarray*} \dot{\theta}(t) & = & H(\theta)^{-1}\,\omega^{b}(t)\\ -\dot{p}(t) & = & R_{0}^{T}\, V_{0}+v(t)\\ -\dot{v}(t) & = & R_{0}^{T}\, g+R_{b}^{0}(t)a^{b}(t) +\dot{p}(t) & = & R_{0}^{T}\,V_{0}+v(t)\\ +\dot{v}(t) & = & R_{0}^{T}\,g+R_{b}^{0}(t)a^{b}(t) \end{eqnarray*} \end_inset @@ -1015,7 +1033,7 @@ v(t)=v_{g}(t)+v_{a}(t) evolving as \begin_inset Formula \begin{eqnarray*} -\dot{v}_{g}(t) & = & R_{i}^{T}\, g\\ +\dot{v}_{g}(t) & = & R_{i}^{T}\,g\\ \dot{v}_{a}(t) & = & R_{b}^{i}(t)a^{b}(t) \end{eqnarray*} @@ -1041,7 +1059,7 @@ p(t)=p_{i}(t)+p_{g}(t)+p_{v}(t) evolving as \begin_inset Formula \begin{eqnarray*} -\dot{p}_{i}(t) & = & R_{i}^{T}\, V_{i}\\ +\dot{p}_{i}(t) & = & R_{i}^{T}\,V_{i}\\ \dot{p}_{g}(t) & = & v_{g}(t)=R_{i}^{T}gt\\ \dot{p}_{v}(t) & = & v_{a}(t) \end{eqnarray*} @@ -1096,7 +1114,7 @@ Predict the NavState from \begin_inset Formula \[ -X_{j}=\mathcal{R}_{X_{i}}(\zeta(t_{ij}))=\left\{ \Phi_{R_{0}}\left(\theta(t_{ij})\right),P_{i}+V_{i}t_{ij}+\frac{gt_{ij}^{2}}{2}+R_{i}\, p_{v}(t_{ij}),V_{i}+gt_{ij}+R_{i}\, v_{a}(t_{ij})\right\} +X_{j}=\mathcal{R}_{X_{i}}(\zeta(t_{ij}))=\left\{ \Phi_{R_{0}}\left(\theta(t_{ij})\right),P_{i}+V_{i}t_{ij}+\frac{gt_{ij}^{2}}{2}+R_{i}\,p_{v}(t_{ij}),V_{i}+gt_{ij}+R_{i}\,v_{a}(t_{ij})\right\} \] \end_inset @@ -1372,7 +1390,7 @@ B_{k}=\left[\begin{array}{c} 0_{3\times3}\\ R_{k}\frac{\Delta_{t}}{2}^{2}\\ R_{k}\Delta_{t} -\end{array}\right],\,\,\,\, C_{k}=\left[\begin{array}{c} +\end{array}\right],\,\,\,\,C_{k}=\left[\begin{array}{c} H(\theta_{k})^{-1}\Delta_{t}\\ 0_{3\times3}\\ 0_{3\times3} @@ -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 diff --git a/doc/ImuFactor.pdf b/doc/ImuFactor.pdf index 0b13c1f5948de38aa6454732ca20367cc630c62c..37badae9cc979302341b1dbac23e03b5be7445d9 100644 GIT binary patch delta 155716 zcmZs?Q*b71(6t-ewv8t?C$>59#J2Iowr$(CZQIVo6DRxq{#}3VgKu};Ro%z;(Y1Q5 zPLD>Y*++~ggJ5N0CP`{opaC+*&d5NppakDOV)!OP$jP~X$4&C)?@yeI^NYp6&CUXh zObqshuS>e%EBh+5*!5p_Ri0m#AP+X0sg!A*uE3M^5v@6XlBs8R z!iV=Nv)%48!H2~pVf+Y2xp+&C9buu=BaMGMPov^lC9%M#H1cSSgN`zAHP*`Tp0azD zw$VYn#MtTlFTS!Z_AgMV1#tv8Q#%u97bjCA+y9yD|5(AXu#hm5{LjG8562{DYG>|Z zLBht$!bb96CABP20B`WE?YG4|sJiKOAToI$L*p6-(pUm8M2GITLi 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