Noise propagation

release/4.3a0
Frank Dellaert 2016-01-17 19:18:20 -08:00
parent 43520265aa
commit be658119b9
1 changed files with 16 additions and 315 deletions

View File

@ -1196,7 +1196,7 @@ Even when we assume uncorrelated noise on
\begin_inset Formula $\theta_{k}$
\end_inset
and
and
\begin_inset Formula $v_{k}$
\end_inset
@ -1213,7 +1213,7 @@ reference "eq:euler_theta"
\end_inset
-
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:euler_v"
@ -1227,7 +1227,7 @@ reference "eq:euler_v"
\begin_inset Formula
\[
\zeta_{k+1}=f\left(\zeta_{k},\omega_{k}^{b},a_{k}^{b}\right)
\zeta_{k+1}=f\left(\zeta_{k},a_{k}^{b},\omega_{k}^{b}\right)
\]
\end_inset
@ -1283,6 +1283,15 @@ where
.
\end_layout
\begin_layout Standard
We start with the noise propagation on
\begin_inset Formula $\theta$
\end_inset
, which is independent of the other quantities.
Taking the derivative, we have
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
@ -1314,7 +1323,7 @@ For the derivatives of
we need the derivative
\begin_inset Formula
\[
\deriv{R_{k}a_{k}^{b}}{\theta_{k}}=-R_{k}\Skew{a_{k}^{b}}\deriv{R_{k}}{\theta_{k}}=-R_{k}\Skew{a_{k}^{b}}H(\theta_{k})
\deriv{R_{k}a_{k}^{b}}{\theta_{k}}=R_{k}\Skew{-a_{k}^{b}}\deriv{R_{k}}{\theta_{k}}=R_{k}\Skew{-a_{k}^{b}}H(\theta_{k})
\]
\end_inset
@ -1322,7 +1331,7 @@ For the derivatives of
where we used
\begin_inset Formula
\[
\deriv{\left(Ra\right)}R\approx-R\Skew a
\deriv{\left(Ra\right)}R\approx R\Skew{-a}
\]
\end_inset
@ -1349,8 +1358,8 @@ Putting all this together, we finally obtain
\[
A_{k}\approx\left[\begin{array}{ccc}
I_{3\times3}-\frac{\Delta_{t}}{2}\Skew{\omega_{k}^{b}}\\
-R_{k}\Skew{a_{k}^{b}}H(\theta_{k})\frac{\Delta_{t}}{2}^{2} & I_{3\times3} & I_{3\times3}\Delta_{t}\\
-R_{k}\Skew{a_{k}^{b}}H(\theta_{k})\Delta_{t} & & I_{3\times3}
R_{k}\Skew{-a_{k}^{b}}H(\theta_{k})\frac{\Delta_{t}}{2}^{2} & I_{3\times3} & I_{3\times3}\Delta_{t}\\
R_{k}\Skew{-a_{k}^{b}}H(\theta_{k})\Delta_{t} & & I_{3\times3}
\end{array}\right]
\]
@ -1373,314 +1382,6 @@ H(\theta_{k})^{-1}\Delta_{t}\\
\end_inset
\end_layout
\begin_layout Standard
A more accurate partial derivative of
\begin_inset Formula $H(\theta_{k})^{-1}$
\end_inset
can be used, as well.
\end_layout
\begin_layout Section
Old Stuff:
\end_layout
\begin_layout Standard
We only measure
\begin_inset Formula $\omega$
\end_inset
and
\begin_inset Formula $a$
\end_inset
at discrete instants of time, and hence we need to make choices about how
to integrate the equations above numerically.
For a vehicle such as a quadrotor UAV, it is not a bad assumption to model
\begin_inset Formula $\omega$
\end_inset
and
\begin_inset Formula $a$
\end_inset
as piecewise constant in the body frame, as the actuation is fixed to the
body.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
The group operation inherited from
\begin_inset Formula $GL(7)$
\end_inset
yields the following result,
\begin_inset Formula
\[
\left[\begin{array}{ccc}
R_{1} & & p_{1}\\
& R_{1} & v_{1}\\
& & 1
\end{array}\right]\left[\begin{array}{ccc}
R_{2} & & p_{2}\\
& R_{2} & v_{2}\\
& & 1
\end{array}\right]=\left[\begin{array}{ccc}
R_{1}R_{2} & & p_{1}+R_{1}p_{2}\\
& R_{1}R_{2} & v_{1}+R_{1}v_{2}\\
& & 1
\end{array}\right]
\]
\end_inset
i.e.,
\begin_inset Formula $R_{2}$
\end_inset
,
\begin_inset Formula $p_{2}$
\end_inset
, and
\begin_inset Formula $v_{2}$
\end_inset
are to interpreted as quantities in the body frame.
How can we achieve a constant angular velocity/constant acceleration operation
with this representation? For an infinitesimal interval
\begin_inset Formula $\delta$
\end_inset
, we expect the result to be
\begin_inset Formula
\[
\left[\begin{array}{ccc}
R+R\hat{\omega}\delta & & p+v\delta\\
& R+R\hat{\omega}\delta & v+Ra\delta\\
& & 1
\end{array}\right]
\]
\end_inset
This can NOT be achieved by
\begin_inset Formula
\[
\left[\begin{array}{ccc}
R & & p\\
& R & v\\
& & 1
\end{array}\right]\left[\begin{array}{ccc}
I+\hat{\omega}\delta & \delta & 0\\
& I+\hat{\omega}\delta & a\delta\\
& & 1
\end{array}\right]
\]
\end_inset
because it is not closed.
Hence, the exponential map as defined below does not really do it for us
\begin_inset Formula
\[
\left[\begin{array}{ccc}
R & & p\\
& R & v\\
& & 1
\end{array}\right]=\lim_{n\rightarrow\infty}\left(I+\left[\begin{array}{ccc}
\hat{\omega} & & v^{b}\\
& \hat{\omega} & a\\
& & 1
\end{array}\right]\frac{\Delta t}{n}\right)^{n}=\left[\begin{array}{ccc}
R+R\hat{\omega}\delta & & p+v\delta\\
& R+R\hat{\omega}\delta & v+Ra\delta\\
& & 1
\end{array}\right]
\]
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
For a quadrotor, forces induced by wind effects and drag are arguably better
approximated over short intervals as constant in the navigation frame.
\end_layout
\begin_layout Standard
Let us examine what we obtain using a constant angular velocity in the body,
but with a zero-order hold on acceleration in the navigation frame instead.
While
\begin_inset Formula $a$
\end_inset
is still measured in the body frame, we rotate it once by
\begin_inset Formula $R_{0}$
\end_inset
at
\begin_inset Formula $t=0$
\end_inset
, and we obtain a much simplified picture:
\begin_inset Formula
\begin{eqnarray*}
R(t) & = & R_{0}\exp\hat{\omega}t\\
v(t) & = & v_{0}+\left(g+R_{0}a\right)t
\end{eqnarray*}
\end_inset
Plugging this into position now yields a much simpler equation, as well:
\begin_inset Formula
\begin{eqnarray*}
p(t) & = & p_{0}+v_{0}t+\left(g+R_{0}a\right)\frac{t^{2}}{2}
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
The goal of the IMU factor is to integrate IMU measurements between two
successive frames and create a binary factor between two NavStates.
Integrating successive gyro and accelerometer readings using the above
equations over each constant interval yields
\begin_inset Formula
\begin{eqnarray}
R_{k+1} & = & R_{k}\exp\hat{\omega}_{k}\Delta t_{k}\label{eq:iter_Rk}\\
p_{k+1} & = & p_{k}+v_{k}\Delta t_{k}+\left(g+R_{k}a_{k}\right)\frac{\left(\Delta t_{k}\right)^{2}}{2}=p_{k}+\frac{v_{k}+v_{k+1}}{2}\Delta t_{k}\nonumber \\
v_{k+1} & = & v_{k}+(g+R_{k}a_{k})\Delta t_{k}\nonumber
\end{eqnarray}
\end_inset
Starting
\begin_inset Formula $X_{i}=(R_{i},p_{i},v_{i})$
\end_inset
, we then obtain an expression for
\begin_inset Formula $X_{j}$
\end_inset
,
\begin_inset Formula
\begin{eqnarray*}
R_{j} & = & R_{i}\prod_{k}\exp\hat{\omega}_{k}\Delta t_{k}\\
p_{j} & = & p_{i}+\sum_{k}v_{k}\Delta t_{k}+\sum_{k}\left(g+R_{k}a_{k}\right)\frac{\left(\Delta t_{k}\right)^{2}}{2}\\
v_{j} & = & v_{i}+\sum_{k}(g+R_{k}a_{k})\Delta t_{k}
\end{eqnarray*}
\end_inset
where
\begin_inset Formula $R_{k}$
\end_inset
has to be updated as in
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:iter_Rk"
\end_inset
.
Note that
\begin_inset Formula
\[
v_{k}=v_{i}+\sum_{l}(g+R_{l}a_{l})\Delta t_{l}
\]
\end_inset
and hence
\begin_inset Formula
\[
p_{j}=p_{i}+\sum_{k}\left(v_{i}+\sum_{l}(g+R_{l}a_{l})\Delta t_{l}\right)\Delta t_{k}+\sum_{k}\left(g+R_{k}a_{k}\right)\frac{\left(\Delta t_{k}\right)^{2}}{2}
\]
\end_inset
\end_layout
\begin_layout Standard
[Is there not a 3/2 power here as in the RSS paper?]
\end_layout
\begin_layout Standard
A crucial problem here is that we depend on a good estimate of
\begin_inset Formula $R_{k}$
\end_inset
at all times, which includes the potentially wrong estimate for the initial
attitude
\begin_inset Formula $R_{i}$
\end_inset
.
\end_layout
\begin_layout Standard
The idea behind the preintegrated IMU factor is two-fold: (a) treat gravity
separately, in the navigation frame, and (b) instead of integrating in
absolute coordinates, we want the IMU integration to happen in the frame
\begin_inset Formula $(R_{i},p_{i},v_{i})$
\end_inset
.
The first idea is easily incorporated by separating out all gravity-related
components:
\begin_inset Formula
\begin{eqnarray*}
p_{j} & = & p_{i}+\sum_{k}\left(\sum_{l}g\Delta t_{l}\right)\Delta t_{k}+\sum_{k}\left(v_{i}+\sum_{l}R_{l}a_{l}\Delta t_{l}\right)\Delta t_{k}+\sum_{k}g\frac{\left(\Delta t_{k}\right)^{2}}{2}+\sum_{k}R_{k}a_{k}\frac{\left(\Delta t_{k}\right)^{2}}{2}\\
v_{j} & = & v_{i}+g\sum_{k}\Delta t_{k}+\sum_{k}R_{k}a_{k}\Delta t_{k}
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
The binary factor is set up as a prediction:
\begin_inset Formula
\[
X_{j}\approx X_{i}\oplus dX_{ij}
\]
\end_inset
where
\begin_inset Formula $dX_{ij}$
\end_inset
is a tangent vector in the tangent space
\begin_inset Formula $T_{i}$
\end_inset
to the manifold at
\begin_inset Formula $X_{i}$
\end_inset
.
\end_layout
\begin_layout Standard