Last updated: 2021-06-26

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Knit directory: VSA_altitude_hold/

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Rmd	d6f3405	Ross Gayler	2021-06-26	wflow_publish("analysis/*.Rmd", republish = TRUE)
Rmd	c39cb39	Ross Gayler	2021-06-25	Take first look at simulation data
html	c39cb39	Ross Gayler	2021-06-25	Take first look at simulation data

This notebook characterises data generated by the simulation of the classically implemented altitude hold controller. We need to understand the properties of the signals in order to design VSA implementations of them.

The values to be characterised are all the real scalars corresponding to the nodes in the data flow diagram for Design 01.

1 Data

The data is generated from runs of the classically implemented simulation.

The data files are stored in the data directory.

The data is supplied as CSV files, each file corresponding to a different run of the simulation.

Each run of the simulation corresponds to a set of starting conditions and parameters.

The only starting condition that varies between runs is the initial altitude. In effect, the multicopter is dropped from some height, with the motors stopped and the drone must command the motors to attain the target altitude.

The name of each data file contains the values of all the parameters. In this work the target altitude is always constant at 5 metres, but more generally this should be treated as a demand which can vary with time.

Each row of the data file corresponds to a point in time and successive rows correspond to successive points in time. (This is a discrete time simulation with fixed time steps.)

Each file contains 500 time steps.

Each columns of the data file corresponds to a node of the data flow diagram.

Only a subset of the nodes are supplied (the nodes in rectangular boxes in the DFD supplied by Simon Levy) and the values of the other nodes can be reconstructed mathematically.

The values supplied from the input files are recorded to three decimal places, so there is scope for approximation error due to the limited precision.

Where a node value is supplied from the input file use that value and can also be mathematically reconstructed from upstream values, us the input value (rather than the reconstructed value) as input to downstream calculations. This is to avoid propagating approximation errors.

1.1 Read data

Read the data from the simulations and mathematically reconstruct the values of the nodes not included in the input files.

# function to clip value to a range
clip <- function(
  x, # numeric
  x_min, # numeric[1] - minimum output value
  x_max  # numeric[1] - maximum output value
) # value # numeric - x constrrained to the range [x_min, x_max]
  {
  x %>% pmax(x_min) %>% pmin(x_max)
}

# function to extract a parametr value from the file name
get_param <- function(
  file, # character - vector of file names
  regexp # character[1] - regular expression for "param=value"
         # use a capture group to get the value part
) # value # numeric - vector of parameter values
{
  file %>% str_match(regexp) %>% 
      subset(select = 2) %>% as.numeric()
}

# read the data
d_wide <- fs::dir_ls(path = here::here("data"), regexp = "/k_start=.*\\.csv$") %>% # get file paths
  vroom::vroom(id = "file") %>% # read files
  dplyr::mutate( # add extra columns
    file = file %>% fs::path_ext_remove() %>% fs::path_file(), # get file name
    # get parameters
    k_start  = file %>% get_param("k_start=([.0-9]+)"), 
    k_p      = file %>% get_param("kp=([.0-9]+)"),
    k_i      = file %>% get_param("Ki=([.0-9]+)"),
    k_tgt    = file %>% get_param("k_tgt=([.0-9]+)"),
    k_windup = file %>% get_param("k_windup=([.0-9]+)"),
    # reconstruct the missing nodes
    i1 = k_tgt - z,
    i2 = i1 - dz,
    i3 = e * k_p,
    i9 = lag(ei, n = 1, default = 0),
    i4 = e + i9,
    i5 = i4 %>% clip(-k_windup, k_windup),
    i6 = ei * k_i,
    i7 = i3 + i6,
    i8 = i7 %>% clip(0, 1)
  ) %>% 
  # add time variable per file
  dplyr::group_by(file) %>% 
  dplyr::mutate(t = 1:n()) %>% 
  dplyr::ungroup()

Rows: 1500 Columns: 6

── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (5): z, dz, e, ei, u


ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

dplyr::glimpse(d_wide)

Rows: 1,500
Columns: 21
$ file     <chr> "k_start=10.00_k_tgt=5.00_kp=0.20_Ki=3.00_k_windup=0.20", "k_…
$ z        <dbl> 10.000, 9.999, 9.998, 9.995, 9.992, 9.987, 9.982, 9.975, 9.96…
$ dz       <dbl> -0.058, -0.156, -0.254, -0.352, -0.450, -0.548, -0.646, -0.74…
$ e        <dbl> -4.942, -4.843, -4.744, -4.643, -4.542, -4.439, -4.335, -4.23…
$ ei       <dbl> -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -…
$ u        <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ k_start  <dbl> 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1…
$ k_p      <dbl> 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0…
$ k_i      <dbl> 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3…
$ k_tgt    <dbl> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5…
$ k_windup <dbl> 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0…
$ i1       <dbl> -5.000, -4.999, -4.998, -4.995, -4.992, -4.987, -4.982, -4.97…
$ i2       <dbl> -4.942, -4.843, -4.744, -4.643, -4.542, -4.439, -4.336, -4.23…
$ i3       <dbl> -0.9884, -0.9686, -0.9488, -0.9286, -0.9084, -0.8878, -0.8670…
$ i9       <dbl> 0.0, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0…
$ i4       <dbl> -4.942, -5.043, -4.944, -4.843, -4.742, -4.639, -4.535, -4.43…
$ i5       <dbl> -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -0.2, -…
$ i6       <dbl> -0.6, -0.6, -0.6, -0.6, -0.6, -0.6, -0.6, -0.6, -0.6, -0.6, -…
$ i7       <dbl> -1.5884, -1.5686, -1.5488, -1.5286, -1.5084, -1.4878, -1.4670…
$ i8       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ t        <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18…

1.2 Check data

Check the data for consistency.

1.2.1 Initial values

The different starting conditions correspond to the multicopter being dropped/released at different altitudes. At the moment of release the multicopter should be at exactly the starting altitude and have zero vertical velocity. Check that this is true.

The initial vertical velocities (\(dz\)) are nonzero. The most likely interpretation is that the \(z\) and \(dz\) values are skewed by one time step. An alternative interpretation is that the multicopter was released one time step earlier and a little higher than the target altitude so that it passes through the target altitude exactly one time step after release.

This apparent skew probably doesn’t make any substantial difference to the qualitative performance of the system. However, it makes some of the graphs messier because the trajectory doesn’t start at zero. It may also lead to the altitude converging slightly off the target altitude because the “perceived” altitude is not quite time-aligned with the error signal.

1.2.2 Check \(z\) and \(dz\)

The altitude (\(z\)) and vertical velocity (\(dz\)) are input values fro the simulation data files. Given only a single point in time, they are independent variables. However, we are given a time series of values and the vertical velocity should be a function of the time series of altitude.

Given that the multicopter simulation is a discrete time simulation, the most likely interpretation is that \(dz\) is calculated as the difference between successive \(z\) values. Check whether that is the case.

Warning: Removed 3 rows containing missing values (geom_point).

The supplied \(dz\) is a linear transform of estimated vertical velocity (difference between successive altitude values).
- I am prepared to believe that the inacurracy is due to the low precision of the values in the files.
But, the scaling is very different: \(dz = 100 dz_{est}\). The time step is 10 ms, so it looks like like \(dz\) is in units of metres per second and \(dz_{est}\) is in units of metres per time step.

Looking at the PID calculation of error (\(e\)): \[ \begin{aligned} e &= k_{tgt} - z - dz \\ &= k_{tgt} - (z + dz) \end{aligned} \]
The term \((z + dz)\) is effectively a prediction of the altitude at the next time step based on the altitude \(z\) and vertical velocity \(dz\) at the previous time step. Thus, the error \(e\) is the difference between the target altitude and the predicted altitude. This requires \(z\) and \(dz\) to be measured in compatible units for the prediction \((z + dz)\) to make sense. Both quantities must be measured in metres and \(dz\) must be metres over the relevant time period, which is one time step.

Consequently, I believe \(dz\) is scaled wrongly because the prediction \((z + dz)\) is the prediction for a point 100 time steps in the future.

1.2.3 Check calculated values

Where possible compare the calculated values with the values recorded from the simulator.

Nodes with a single input edge can be checked against the output of the predecessor node.

The nodes which can be checked are: e, ei, u

      Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
-1.000e-03  0.000e+00  0.000e+00  1.733e-05  0.000e+00  1.000e-03

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 -1e-03   0e+00   0e+00  -1e-05   0e+00   1e-03

      Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
-2.000e-03 -8.000e-04  0.000e+00  3.627e-05  1.000e-03  2.000e-03

The reconstructed values which can be checked against values from the simulation are correct except for approximation error due to the low precision of the numbers in the files.

2 Distribution summary

Display a quick summary of the distributions of values of all the nodes.

Note that these summaries are over all the input files pooled.

Table 2.1: Data summary
Name	d_wide
Number of rows	1500
Number of columns	21
_______________________
Column type frequency:
character	1
numeric	20
________________________
Group variables	None

Variable type: character

skim_variable	n_missing	complete_rate	min	max	empty	n_unique	whitespace
file	0	1	53	54	0	3	0

Variable type: numeric

skim_variable	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
z	1	5.41	1.32	2.00	4.92	5.11	5.70	10.00	▁▇▅▁▁
dz	1	-0.33	1.10	-4.37	-0.66	-0.11	0.07	2.70	▁▂▆▇▁
e	1	-0.08	0.57	-4.94	0.00	0.00	0.00	3.27	▁▁▁▇▁
ei	1	0.16	0.08	-0.20	0.17	0.17	0.18	0.20	▁▁▁▁▇
u	1	0.51	0.12	0.00	0.52	0.53	0.54	1.00	▁▁▇▁▁
k_start	1	6.67	3.40	2.00	2.00	8.00	10.00	10.00	▇▁▁▇▇
k_p	1	0.20	0.00	0.20	0.20	0.20	0.20	0.20	▁▁▇▁▁
k_i	1	3.00	0.00	3.00	3.00	3.00	3.00	3.00	▁▁▇▁▁
k_tgt	1	5.00	0.00	5.00	5.00	5.00	5.00	5.00	▁▁▇▁▁
k_windup	1	0.20	0.00	0.20	0.20	0.20	0.20	0.20	▁▁▇▁▁
i1	1	-0.41	1.32	-5.00	-0.70	-0.11	0.08	3.00	▁▁▅▇▁
i2	1	-0.08	0.57	-4.94	0.00	0.00	0.00	3.27	▁▁▁▇▁
i3	1	-0.02	0.11	-0.99	0.00	0.00	0.00	0.65	▁▁▁▇▁
i9	1	0.16	0.08	-0.20	0.17	0.17	0.18	0.20	▁▁▁▁▇
i4	1	0.08	0.64	-5.04	0.17	0.17	0.18	3.44	▁▁▁▇▁
i5	1	0.16	0.08	-0.20	0.17	0.17	0.18	0.20	▁▁▁▁▇
i6	1	0.48	0.24	-0.60	0.52	0.52	0.54	0.60	▁▁▁▁▇
i7	1	0.46	0.34	-1.59	0.52	0.52	0.54	1.25	▁▁▁▇▁
i8	1	0.51	0.12	0.00	0.52	0.52	0.54	1.00	▁▁▇▁▁
t	1	250.50	144.39	1.00	125.75	250.50	375.25	500.00	▇▇▇▇▇

3 Plots

Plot the relationships between the major nodes. This is not necessarily of immediate use in deciding the design fo VSA components, but gives a feel for the dynamics of the system.

3.1 Time explicit

Show the values of the major nodes as a function of time.

Warning: Removed 1 row(s) containing missing values (geom_path).

3.2 Time implicit

Show the values of the major nodes as a function of each other (with time implicit).

Warning: Removed 1 row(s) containing missing values (geom_path).

Warning: Removed 1 row(s) containing missing values (geom_path).

Warning: Removed 1 row(s) containing missing values (geom_path).

Warning: Removed 1 row(s) containing missing values (geom_path).

Warning: Removed 1 row(s) containing missing values (geom_path).

R version 4.1.0 (2021-05-18)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 21.04

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.9.0
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.9.0

locale:
 [1] LC_CTYPE=en_AU.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_AU.UTF-8        LC_COLLATE=en_AU.UTF-8    
 [5] LC_MONETARY=en_AU.UTF-8    LC_MESSAGES=en_AU.UTF-8   
 [7] LC_PAPER=en_AU.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_AU.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices datasets  utils     methods   base     

other attached packages:
[1] ggplot2_3.3.4 forcats_0.5.1 skimr_2.1.3   stringr_1.4.0 DT_0.18      
[6] dplyr_1.0.7   vroom_1.5.1   here_1.0.1    fs_1.5.0     

loaded via a namespace (and not attached):
 [1] tidyselect_1.1.1  xfun_0.24         repr_1.1.3        purrr_0.3.4      
 [5] colorspace_2.0-2  vctrs_0.3.8       generics_0.1.0    htmltools_0.5.1.1
 [9] yaml_2.2.1        base64enc_0.1-3   utf8_1.2.1        rlang_0.4.11     
[13] later_1.2.0       pillar_1.6.1      glue_1.4.2        withr_2.4.2      
[17] bit64_4.0.5       lifecycle_1.0.0   munsell_0.5.0     gtable_0.3.0     
[21] workflowr_1.6.2   htmlwidgets_1.5.3 evaluate_0.14     labeling_0.4.2   
[25] knitr_1.33        tzdb_0.1.1        crosstalk_1.1.1   httpuv_1.6.1     
[29] parallel_4.1.0    fansi_0.5.0       highr_0.9         Rcpp_1.0.6       
[33] renv_0.13.2       promises_1.2.0.1  scales_1.1.1      jsonlite_1.7.2   
[37] farver_2.1.0      bit_4.0.4         digest_0.6.27     stringi_1.6.2    
[41] bookdown_0.22     rprojroot_2.0.2   grid_4.1.0        cli_2.5.0        
[45] tools_4.1.0       magrittr_2.0.1    tibble_3.1.2      tidyr_1.1.3      
[49] crayon_1.4.1      whisker_0.4       pkgconfig_2.0.3   ellipsis_0.3.2   
[53] rmarkdown_2.9     rstudioapi_0.13   R6_2.5.0          git2r_0.28.0     
[57] compiler_4.1.0

Characterise Simulation Data

Ross Gayler

2021-06-22