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docs: Fix latex formatting, and ignore file #297

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3 changes: 2 additions & 1 deletion .pre-commit-config.yaml
Original file line number Diff line number Diff line change
Expand Up @@ -2,7 +2,8 @@ exclude: |
(?x)^(
docs/conf.py |
.devcontainer/ |
tests/ops_data.py
tests/ops_data.py |
docs/gnssa_preprocessing.md
)
ci:
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26 changes: 13 additions & 13 deletions docs/gnssa_preprocessing.md
Original file line number Diff line number Diff line change
Expand Up @@ -64,11 +64,11 @@ based upon an estimated state $Y_{(k-1|k-1)}$ at a previous time $k-1$.
The forward filter first uses the previous measurement update, $Y_{(k-1|k-1)}$
and $P_{(k-1|k-1)}$, to predict the state vector and covariance:

$$ Y*{(k|k-1)} = \Phi*{(k|k-1)} \cdot Y\_{(k-1|k-1)} $$
$$ Y_{(k|k-1)} = \Phi_{(k|k-1)} \cdot Y_{(k-1|k-1)} $$

$$
P*{(k|k-1)} = \Phi*{(k|k-1)} \cdot P*{(k-1|k-1)} \cdot \Phi^T*{(k|k-1)} +
Q\_{(k|k-1)}
P_{(k|k-1)} = \Phi_{(k|k-1)} \cdot P_{(k-1|k-1)} \cdot \Phi^T_{(k|k-1)} +
Q_{(k|k-1)}
$$

In the above equations, $\Phi_{(k|k-1)}$ describes the physics of the system
Expand All @@ -77,8 +77,8 @@ and $t_{k-1}$) and $Q_{(k|k-1)}$ is a noise parameter associated with
$\Phi_{(k|k-1)}$. Based upon this prediction, we calculate the Kalman gain:

$$
K*{(k|k)} = P*{(k|k-1)} \cdot H^T*{(k|k)} \cdot \left( H*{(k|k)} \cdot
P*{(k|k-1)} \cdot H^T*{(k|k)} + V\_{(k|k)} \right)^{-1}
K_{(k|k)} = P_{(k|k-1)} \cdot H^T_{(k|k)} \cdot \left( H_{(k|k)} \cdot
P_{(k|k-1)} \cdot H^T_{(k|k)} + V_{(k|k)} \right)^{-1}
$$

$V_{(k|k)}$ is the measurement covariance at time $t_k$ and $H_{(k|k)}$ is a
Expand All @@ -87,11 +87,11 @@ measurements $z_k$ at time $t_k$. From this, the measurement update of $Y$ and
$P$ is calculated as:

$$
Y*{(k|k)} = Y*{(k|k-1)} + K*{(k|k)} \cdot \left( z*{(k|k)} - H*{(k|k)} \cdot
Y*{(k|k-1)} \right)
Y_{(k|k)} = Y_{(k|k-1)} + K_{(k|k)} \cdot \left( z_{(k|k)} - H_{(k|k)} \cdot
Y_{(k|k-1)} \right)
$$

$$ P*{(k|k)} = \left( I - K*{(k|k)} \cdot H*{(k|k)} \right) \cdot P*{(k|k-1)} $$
$$ P_{(k|k)} = \left( I - K_{(k|k)} \cdot H_{(k|k)} \right) \cdot P_{(k|k-1)} $$

Note that $H_{(k|k)}=0$ during ping reply epochs since no new positions or
velocities are collected at those times. Thus, there is no measurement update at
Expand All @@ -101,16 +101,16 @@ The back-smoother is a Rauch-Tung-Striebel smoother (Rauch et al., 1965) that
uses a smoothing gain $A_{(k)}$ to update $Y$ and $P$ based upon future
measurement updates calculated by the forward filter:

$$ A*{(k)} = P*{(k|k)} \cdot \Phi^T*{(k+1|k)} \cdot P^{-1}*{(k+1|k)} $$
$$ A_{(k)} = P_{(k|k)} \cdot \Phi^T_{(k+1|k)} \cdot P^{-1}_{(k+1|k)} $$

$$
Y*{(k|N)} = Y*{(k|k)} + A*{(k)} \cdot \left( Y*{(k+1|N)} - Y\_{(k+1|k)}
Y_{(k|N)} = Y_{(k|k)} + A_{(k)} \cdot \left( Y_{(k+1|N)} - Y_{(k+1|k)}
\right)
$$

$$
P*{(k|N)} = P*{(k|k)} + A*{(k)} \cdot \left( P*{(k+1|N)} - P*{(k+1|k)}
\right) \cdot A*{(k)}^T
P_{(k|N)} = P_{(k|k)} + A_{(k)} \cdot \left( P_{(k+1|N)} - P_{(k+1|k)}
\right) \cdot A_{(k)}^T
$$

## Antenna-Transducer Rotation
Expand Down Expand Up @@ -141,7 +141,7 @@ transducer position may be computed by rotating $X_{body}$ about the three
body-frame axes:

$$
X*{trans} = X*{ant} + \mathbf{R}_3(h-360) \mathbf{R}\_2(-p) \mathbf{R}\_1(-r)
X_{trans} = X_{ant} + \mathbf{R}_3(h-360) \mathbf{R}_2(-p) \mathbf{R}_1(-r)
X_{body}
$$

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