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fix small typos

release/4.3a0
Duy-Nguyen Ta 2015-12-22 00:28:04 -05:00
parent a6cc7ef2dc
commit e2dbfa1b12
1 changed files with 11 additions and 11 deletions
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20
doc/ImuFactor.lyx
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@ -168,7 +168,7 @@ vector field
\end_inset
, defined as a mapping from
\begin_inset Formula $R\times M$
\begin_inset Formula $\Rone\times M$
\end_inset
to tangent vectors at
@ -245,7 +245,7 @@ X(t)=\left\{ R_{0},P_{0}+V_{0}t,V_{0}\right\}
then the differential equation describing the trajectory is
\begin_inset Formula
\[
X'(t)=\left[0_{3x3},V_{0},0_{3x1}\right],\,\,\,\,\, X(0)=\left\{ R_{0},P_{0},V_{0}\right\}
\dot{X}(t)=\left[0_{3x3},V_{0},0_{3x1}\right],\,\,\,\,\,X(0)=\left\{ R_{0},P_{0},V_{0}\right\}
\]
\end_inset
@ -264,7 +264,7 @@ Valid vector fields on a NavState manifold are special, in that the attitude
:
\begin_inset Formula
\begin{equation}
X'(t)=\left[W(X,t),V(t),A(X,t)\right]\label{eq:validField}
\dot{X}(t)=\left[W(X,t),V(t),A(X,t)\right]\label{eq:validField}
\end{equation}
\end_inset
@ -297,7 +297,7 @@ where
, and hence the resulting exact vector field is
\begin_inset Formula
\begin{equation}
X'(t)=\left[W(X,t),V(t),A(X,t)\right]=\left[R(t)\Skew{\omega^{b}},V(t),g+R(t)a^{b}(t)\right]\label{eq:bodyField}
\dot{X}(t)=\left[W(X,t),V(t),A(X,t)\right]=\left[R(t)\Skew{\omega^{b}},V(t),g+R(t)a^{b}(t)\right]\label{eq:bodyField}
\end{equation}
\end_inset
@ -640,7 +640,7 @@ and taking the derivative for
we obtain
\begin_inset Formula
\[
\dot{R}(t)=\frac{d\gamma(\delta)}{d\delta}\biggr\vert_{\delta=0}=\frac{d\Phi_{R(t)}\left(H(\theta)\theta'(t)\delta\right)}{dt}\biggr\vert_{t=0}=R(t)\Skew{H(\theta)\dot{\theta}(t)}
\dot{R}(t)=\frac{d\gamma(\delta)}{d\delta}\biggr\vert_{\delta=0}=\frac{d\Phi_{R(t)}\left(H(\theta)\dot{\theta}(t)\delta\right)}{d\delta}\biggr\vert_{\delta=0}=R(t)\Skew{H(\theta)\dot{\theta}(t)}
\]
\end_inset
@ -782,7 +782,7 @@ Here
is a 9-vector, with respectively angular, position, and velocity components.
The tangent vector at
\begin_inset Formula $R_{0}$
\begin_inset Formula $X_{0}$
\end_inset
is
@ -875,7 +875,7 @@ We can create a trajectory
\begin_inset Formula
\[
\gamma(\delta)=X(t+\delta)=\left\{ \Phi_{R_{0}}\left(\theta(t)+\theta'(t)\delta\right),P_{0}+R_{0}\left\{ p(t)+p'(t)\delta\right\} ,V_{0}+R_{0}\left\{ v(t)+v'(t)\delta\right\} \right\}
\gamma(\delta)=X(t+\delta)=\left\{ \Phi_{R_{0}}\left(\theta(t)+\dot{\theta}(t)\delta\right),P_{0}+R_{0}\left\{ p(t)+\dot{p}(t)\delta\right\} ,V_{0}+R_{0}\left\{ v(t)+\dot{v}(t)\delta\right\} \right\}
\]
\end_inset
@ -887,7 +887,7 @@ and taking the derivative for
we obtain
\begin_inset Formula
\[
\dot{X}(t)=\frac{d\gamma(\delta)}{d\delta}\biggr\vert_{\delta=0}=\left[R(t)\Skew{H(\theta)\theta'(t)},R_{0}\, p'(t),R_{0}\, v'(t)\right]
\dot{X}(t)=\frac{d\gamma(\delta)}{d\delta}\biggr\vert_{\delta=0}=\left[R(t)\Skew{H(\theta)\dot{\theta}(t)},R_{0}\,\dot{p}(t),R_{0}\,\dot{v}(t)\right]
\]
\end_inset
@ -902,7 +902,7 @@ reference "eq:bodyField"
, we have exact integration iff
\begin_inset Formula
\[
\left[R(t)\Skew{H(\theta)\theta'(t)},R_{0}\, p'(t),R_{0}\, v'(t)\right]=\left[R(t)\Skew{\omega^{b}},V(t),g+R(t)a^{b}(t)\right]
\left[R(t)\Skew{H(\theta)\dot{\theta}(t)},R_{0}\,\dot{p}(t),R_{0}\,\dot{v}(t)\right]=\left[R(t)\Skew{\omega^{b}},V(t),g+R(t)a^{b}(t)\right]
\]
\end_inset
@ -1036,7 +1036,7 @@ The recipe for the IMU factor is then, in summary.
Solve the ordinary differential equation
\begin_inset Formula
\begin{eqnarray*}
\dot{\theta}(t) & = & H(\theta)^{-1}\,\omega^{b}\\
\dot{\theta}(t) & = & H(\theta(t))^{-1}\,\omega^{b}(t)\\
\dot{p}_{v}(t) & = & v_{a}(t)\\
\dot{v}_{a}(t) & = & R_{b}^{0}(t)a^{b}(t)
\end{eqnarray*}