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Document Kalman filter in the PVT block
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@carlesfernandez
carlesfernandez committed Jul 10, 2023
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nav: "sp-block"
toc: true
toc_sticky: true
last_modified_at: 2023-02-14T09:54:02+02:00
last_modified_at: 2023-07-10T09:54:02+02:00
---

The _PVT_ block is the last one in the GNSS-SDR flow graph. Hence, it acts as a
Expand Down Expand Up @@ -52,7 +52,7 @@ where $$ \mathbf{r}_r $$ is the receiver's antenna position in an
earth-centered, earth-fixed (ECEF) coordinate system (in meters), $$ c $$ is the
speed of light, and $$ dt_r $$ is the receiver clock bias (in seconds).

The measurement vector is defined as:
The measurement vector is defi

$$ \begin{equation} \mathbf{y} = \left(P_r^{(1)}, P_r^{(2)}, P_r^{(3)}, ..., P_r^{(m)} \right)^T~. \end{equation} $$

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- $$ El_r^{(s)} $$ is the elevation angle of satellite direction (in rad).

- $$ \sigma_{bclock,s} = 30 $$ is the standard deviation of the broadcast
clock error (in m).
- $$ \sigma_{bclock,s} $$ is the standard deviation of the broadcast ephemeris
and clock error (in m). This parameter is estimated internally from URA (User
Range Accuracy) or or similar indicators.

- $$ \sigma_{ion,s} $$ is the standard deviation of ionosphere correction
model error (in m). This parameter is set to $$ \sigma_{ion} = 5 $$ m by
Expand Down Expand Up @@ -814,8 +815,6 @@ function](https://gssc.esa.int/navipedia/index.php/Mapping_of_Niell)[^Niell96].
The zenith total delay $$ Z_{T,r} $$ is estimated as an unknown parameter in the
parameter estimation process.



## Estimate the tropospheric zenith total delay and gradient

If the processing option `trop_model` is set to `Estimate_ZTD_Grad`, a more
Expand All @@ -826,14 +825,112 @@ $$ \begin{equation}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!m\left(El_{r}^{(s)}\right)\! = \!m_{W}\left(El_{r}^{(s)}\right)\!\left[1\!+\!\cot \! \left(El_{r}^{(s)}\right) \! \left( \! G_{N,r} \cos \! \left(Az_{r}^{(s)}\right) \!+\! G_{E,r} \sin \! \left(Az_{r}^{(s)}\right)\!\right) \!\right]
\end{equation} $$

where $$ Az_{r}^{(s)} $$ is the azimuth angle of satellite direction (rad), and
$$ G_{E,r} $$ and $$ G_{N,r} $$ are the east and north components of the
where $$ Az_{r}^{(s)} $$ is the azimuth angle of satellite direction (rad), and $$ G_{E,r} $$ and $$ G_{N,r} $$ are the east and north components of the
tropospheric gradient, respectively. The zenith total delay $$ Z_{T,r} $$ and
the gradient parameters $$ G_{E,r} $$ and $$ G_{N,r} $$ are estimated as unknown
parameters in the parameter estimation process.

 

## A de-noising Kalman filter for the PVT solution

The `PVT` block can apply a simple Kalman filter to the computed PVT solutions.
This filter can be enabled by setting `PVT.enable_pvt_kf=true` in the
configuration file. The structure of this filter is as follows:

- **State model:**
$$ \begin{equation}
\mathbf{x} = \left[ x, y, z, v_x, v_y, v_z \right]^{T}
\end{equation} $$
$$ \begin{equation}
\mathbf{x}_k = \mathbf{F} \mathbf{x}_{k-1} + \mathbf{v}_k~, \quad \mathbf{v}_k \sim \mathcal{N}(\mathbf{0},\mathbf{Q})
\end{equation} $$
$$ \begin{equation}
\textbf{F} = \left[ \begin{array}{cccccc}
1 & 0 & 0 & T & 0 & 0 \\
0 & 1 & 0 & 0 & T & 0 \\
0 & 0 & 1 & 0 & 0 & T \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{array} \right]
\end{equation} $$
$$ \begin{equation}
\textbf{Q} = \begin{bmatrix}
\sigma_{s\_pos}^{2} & 0 & 0 & 0 & 0 & 0 \\
0 & \sigma_{s\_pos}^{2} & 0 & 0 & 0 & 0 \\
0 & 0 & \sigma_{s\_pos}^{2} & 0 & 0 & 0 \\
0 & 0 & 0 & \sigma_{s\_vel}^{2} & 0 & 0 \\
0 & 0 & 0 & 0 & \sigma_{s\_vel}^{2} & 0 \\
0 & 0 & 0 & 0 & 0 & \sigma_{s\_vel}^{2}
\end{bmatrix}
\end{equation} $$
- **Measurement model:**
$$ \begin{equation}
\mathbf{z} = \left[ x , y , z , v_{x}, v_{y}, v_{z} \right]^{T}
\end{equation} $$
$$ \begin{equation}
\mathbf{z}_k = \mathbf{H}\mathbf{x}_k + \mathbf{w}_k , \quad \mathbf{w}_k \sim \mathcal{N}(\mathbf{0},\mathbf{R})
\end{equation} $$
$$ \begin{equation} \textbf{H} = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix} \end{equation} $$
$$ \begin{equation} \textbf{R} = \begin{bmatrix}
\sigma_{m\_pos}^{2} & 0 & 0 & 0 & 0 & 0 \\
0 & \sigma_{m\_pos}^{2} & 0 & 0 & 0 & 0 \\
0 & 0 & \sigma_{m\_pos}^{2} & 0 & 0 & 0 \\
0 & 0 & 0 & \sigma_{m\_vel}^{2} & 0 & 0 \\
0 & 0 & 0 & 0 & \sigma_{m\_vel}^{2} & 0 \\
0 & 0 & 0 & 0 & 0 & \sigma_{m\_vel}^{2}
\end{bmatrix} \end{equation} $$

- **Initialization:**
$$ \begin{equation}
\mathbf{x}_{0|0} = \left[ \begin{array}{cccc} x_{0} & y_{0} & z_{0} & v_{x_{0}} & v_{y_{0}} & v_{z_{0}} \end{array} \right]^T
\end{equation} $$

$$\begin{equation}
\mathbf{P}_{0|0} = \begin{bmatrix}
\sigma_{s\_pos}^{2} & 0 & 0 & 0 & 0 & 0 \\
0 & \sigma_{s\_pos}^{2} & 0 & 0 & 0 & 0 \\
0 & 0 & \sigma_{s\_pos}^{2} & 0 & 0 & 0 \\
0 & 0 & 0 & \sigma_{s\_vel}^{2} & 0 & 0 \\
0 & 0 & 0 & 0 & \sigma_{s\_vel}^{2} & 0 \\
0 & 0 & 0 & 0 & 0 & \sigma_{s\_vel}^{2}
\end{bmatrix}
\end{equation} $$

- **Prediction:**
$$ \begin{equation}
\hat{\mathbf{x}}_{k|k-1} = \mathbf{F} \hat{\mathbf{x}}_{k-1|k-1}
\end{equation} $$
$$ \begin{equation}
\mathbf{P}_{k|k-1} = \mathbf{F} \mathbf{P}_{k-1|k-1} \mathbf{F}^T + \mathbf{Q}
\end{equation} $$
- **Update:**
$$ \begin{equation}
\mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{H}^T \left( \mathbf{H}\mathbf{P}_{k|k-1} \mathbf{H}^T + \mathbf{R} \right)^{-1}
\end{equation} $$

$$ \begin{equation}
\hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k \left( \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k|k-1} \right)
\end{equation} $$

$$ \begin{equation}
\mathbf{P}_{k|k} = \left( \mathbf{I} - \mathbf{K}_{k} \mathbf{H} \right)\mathbf{P}_{k|k-1}
\end{equation} $$

The following parameters are exposed in the configuration, here with their defaut values:
$$ \sigma_{m\_pos} = \text{\texttt{PVT.kf\_measures\_ecef\_pos\_sd\_m}} = 1.0 \text {, in [m].} $$
$$ \sigma_{m\_pos} = \text{\texttt{PVT.kf\_measures\_ecef\_vel\_sd\_ms}} = 0.1 \text {, in [m/s].} $$
$$ \sigma_{s\_pos} = \text{\texttt{PVT.kf\_system\_ecef\_pos\_sd\_m}} = 0.01 \text {, in [m].} $$
$$ \sigma_{s\_vel} = \text{\texttt{PVT.kf\_system\_ecef\_vel\_sd\_ms}} = 0.001 \text {, in [m/s].} $$

---------

 
Expand Down Expand Up @@ -1058,6 +1155,11 @@ standard and precise positioning. It accepts the following parameters:
| `enable_protobuf` | [`true`, `false`]: If set to `true`, the data serialization is done using [Protocol Buffers](https://developers.google.com/protocol-buffers/), with the format defined at [`monitor_pvt.proto`](https://github.com/gnss-sdr/gnss-sdr/blob/next/docs/protobuf/monitor_pvt.proto). An example of usage is the [gnss-sdr-monitor](https://github.com/acebrianjuan/gnss-sdr-monitor). If set to `false`, it uses [Boost Serialization](https://www.boost.org/doc/libs/release/libs/serialization/doc/index.html). For an example of usage of the latter, check the [gnss-sdr-pvt-monitoring-client](https://github.com/acebrianjuan/gnss-sdr-pvt-monitoring-client). This parameter defaults to `true` (Protocol Buffers is used). | Optional |
| `use_e6_for_pvt` | [`true`, `false`]: If set to `false`, the PVT engine will ignore observables from Galileo E6B signals. It defaults to `true`, so observables will be used if found. This feature is present in GNSS-SDR v0.0.18 and later versions. | Optional |
| `use_has_corrections` | [`true`, `false`]: If set to `false`, the PVT engine will ignore corrections from the Galileo High Accuracy Service. It defaults to `true`, so corrections will be applied if available. This feature is present in GNSS-SDR v0.0.18 and later versions. | Optional |
| `enable_pvt_kf` | [`true`, `false`]: If set to `true`, it enables the Kalman filter of the PVT solution. It defaults to `false`. <span style="color: orange"> This configuration parameter is only present on the `next` branch of the upstream repository, and it will be included in the next stable release of GNSS-SDR.</span> | Optional |
| `kf_measures_ecef_pos_sd_m` | Standard deviation of the position estimations, in meters. It defaults to `1.0` [m]. Only used if `PVT.enable_pvt_kf=true`. <span style="color: orange"> This configuration parameter is only present on the `next` branch of the upstream repository, and it will be included in the next stable release of GNSS-SDR.</span> | Optional |
| `kf_measures_ecef_vel_sd_ms` | Standard deviation of the velocity estimations, in meters per second. It defaults to `0.1` [m/s]. Only used if `PVT.enable_pvt_kf=true`. <span style="color: orange"> This configuration parameter is only present on the `next` branch of the upstream repository, and it will be included in the next stable release of GNSS-SDR.</span> | Optional |
| `kf_system_ecef_pos_sd_m` | Standard deviation of the dynamic system model for position, in meters. It defaults to `0.01` [m]. Only used if `PVT.enable_pvt_kf=true`. <span style="color: orange"> This configuration parameter is only present on the `next` branch of the upstream repository, and it will be included in the next stable release of GNSS-SDR.</span> | Optional |
| `kf_system_ecef_vel_sd_ms` | Standard deviation of the dynamic system model for velocity, in meters per second. It defaults to `0.001` [m/s]. Only used if `PVT.enable_pvt_kf=true`. <span style="color: orange"> This configuration parameter is only present on the `next` branch of the upstream repository, and it will be included in the next stable release of GNSS-SDR.</span> | Optional |
| `use_unhealthy_sats` | [`true`, `false`]: If set to `true`, the PVT engine will use observables from satellites flagged as unhealthy in the navigation message. It defaults to `false`, so those observables will be ignored. This feature is present in GNSS-SDR v0.0.18 and later versions. | Optional |
| `show_local_time_zone` | [`true`, `false`]: If set to `true`, the time of the PVT solution displayed in the terminal is shown in the local time zone, referred to UTC. It defaults to `false`, so time is shown in UTC. This parameter does not affect time annotations in other output formats, which are always UTC. | Optional |
| `rtk_trace_level` | Configure the RTKLIB trace level (`0`: off, up to `5`: max. verbosity). When set to something > `2`, the RTKLIB library become more verbose in the internal logging file. It defaults to `0` (off). | Optional |
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