---
title: "Compare Models"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Compare Models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

We load the `R` package `navigation`
```{r, message=F, warning=FALSE}
library(navigation)
```

We load the data `lemniscate_traj_ned`
```{r}
data("lemniscate_traj_ned") # trajectory in proper format as shown bellow
head(lemniscate_traj_ned)
```

We make the `trajectory` object
```{r}
traj <- make_trajectory(data = lemniscate_traj_ned, system = "ned")
class(traj)
```
We then define a `timing` object
```{r}
timing <- make_timing(
  nav.start = 0, # time at which to begin filtering
  nav.end = 15,
  freq.imu = 100, # frequency of the IMU, can be slower wrt trajectory frequency
  freq.gps = 1, # gnss frequency
  freq.baro = 1, # barometer frequency (to disable, put it very low, e.g. 1e-5)
  gps.out.start = 8, # to simulate a GNSS outage, set a time before nav.end
  gps.out.end = 13
)
```


We define the sensor model for generating sensor errors.
```{r}
snsr.mdl <- list()
acc.mdl <- WN(sigma2 = 5.989778e-05) + AR1(phi = 9.982454e-01, sigma2 = 1.848297e-10) + AR1(phi = 9.999121e-01, sigma2 = 2.435414e-11) + AR1(phi = 9.999998e-01, sigma2 = 1.026718e-12)
gyr.mdl <- WN(sigma2 = 1.503793e-06) + AR1(phi = 9.968999e-01, sigma2 = 2.428980e-11) + AR1(phi = 9.999001e-01, sigma2 = 1.238142e-12)

snsr.mdl$imu <- make_sensor(name = "imu", frequency = timing$freq.imu, error_model1 = acc.mdl, error_model2 = gyr.mdl)
```

We define the stochastic model for the GPS errors considering a RTK-like GNSS system
```{r}
gps.mdl.pos.hor <- WN(sigma2 = 0.025^2)
gps.mdl.pos.ver <- WN(sigma2 = 0.05^2)
gps.mdl.vel.hor <- WN(sigma2 = 0.01^2)
gps.mdl.vel.ver <- WN(sigma2 = 0.02^2)
snsr.mdl$gps <- make_sensor(
  name = "gps", frequency = timing$freq.gps,
  error_model1 = gps.mdl.pos.hor,
  error_model2 = gps.mdl.pos.ver,
  error_model3 = gps.mdl.vel.hor,
  error_model4 = gps.mdl.vel.ver
)
```

We define the stochastic model for the barometer
```{r}
baro.mdl <- WN(sigma2 = 0.5^2)
snsr.mdl$baro <- make_sensor(name = "baro", frequency = timing$freq.baro, error_model1 = baro.mdl)
```


We then define the stochastic model for the sensor error, here we configure the EKF to have the same model as for data generation (ideal setup).
```{r}
KF.mdl <- list()

KF.mdl$imu <- make_sensor(name = "imu", frequency = timing$freq.imu, error_model1 = acc.mdl, error_model2 = gyr.mdl)
KF.mdl$gps <- snsr.mdl$gps
KF.mdl$baro <- snsr.mdl$baro
```


We then define a wrong model with respect to the data generation process (only composed of the white noise part)

```{r}
wrong_acc.mdl <- WN(sigma2 = 5.989778e-05)
wrong_gyr.mdl <- WN(sigma2 = 1.503793e-06)
wrong_KF.mdl <- list()
wrong_KF.mdl$imu <- make_sensor(name = "imu", frequency = timing$freq.imu, error_model1 = wrong_acc.mdl, error_model2 = wrong_gyr.mdl)
wrong_KF.mdl$gps <- snsr.mdl$gps
wrong_KF.mdl$baro <- snsr.mdl$baro
```


We perform the navigation Monte Carlo simulation considering the correct stochastic model.

```{r, message=FALSE, results='hide', eval=T}
num.runs <- 5 # number of Monte-Carlo simulations
res <- navigation(
  traj.ref = traj,
  timing = timing,
  snsr.mdl = snsr.mdl,
  KF.mdl = KF.mdl,
  num.runs = num.runs,
  noProgressBar = TRUE,
  PhiQ_method = "1",
  parallel.ncores = 1,
  P_subsampling = timing$freq.imu,
  compute_PhiQ_each_n = 50
) # keep one covariance every second
```


We perform the navigation Monte Carlo simulation considering the wrong stochastic model.
```{r, message=FALSE, results='hide', eval=T}
wrong_res <- navigation(
  traj.ref = traj,
  timing = timing,
  snsr.mdl = snsr.mdl,
  KF.mdl = wrong_KF.mdl, # < here the model is the wrong one
  num.runs = num.runs,
  noProgressBar = TRUE,
  PhiQ_method = "1",
  parallel.ncores = 1,
  P_subsampling = timing$freq.imu,
  compute_PhiQ_each_n = 50
) # keep one covariance every second
```



We compute statistics on navigation performance for the navigation simulation that considered the correct stochastic model and for the navigation simulation that considered the wrong stochastic model.
```{r, eval=T}
pe_res <- compute_mean_position_err(res, step = 25)
pe_wrong_res <- compute_mean_position_err(wrong_res, step = 25)

oe_res <- compute_mean_orientation_err(res, step = 25)
oe_wrong_res <- compute_mean_orientation_err(wrong_res, step = 25)
```


```{r, results='hide', fig.align='center', fig.width=7, fig.height=5, eval=T}
nees <- compute_nees(res, step = timing$freq.imu)
wrong_nees <- compute_nees(wrong_res, step = timing$freq.imu)

coverage <- compute_coverage(res, alpha = 0.7, step = timing$freq.imu)
wrong_coverage <- compute_coverage(wrong_res, alpha = 0.7, step = timing$freq.imu)
```



We compare results
```{r,  fig.height=5, fig.align='center', fig.width=6, eval=T}
plot(pe_res, pe_wrong_res, legend = c("correct model", "wrong model"))

plot(oe_res, oe_wrong_res, legend = c("correct model", "wrong model"))

plot(nees, wrong_nees, legend = c("correct model", "wrong model"))

plot(coverage, wrong_coverage, legend = c("correct model", "wrong model"))
```


