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This notebook attempts to statistically characterise the PID controller as a black-box function.

Looking at the data flow diagram for Design 01, and ignoring the tuning parameters, there are three input variables:

  • Altitude (\(z\))
  • Vertical velocity (\(dz\))
  • Target altitude (\(k_{tgt}\))

And one output variable:

  • Clipped motor demand (\(i8 \triangleq u\))

The integrated error nodes (\(i4\), \(i5\), \(i9\)) introduce a history dependence into the PID function. This part of the DFD implements the “I” (integral) term of “PID” controller. However, the previous analyses of the simulation data suggest that the integral term is only effective in the final phase of the approach to the target altitude, and introduces an (at best) harmless oscillation. This suggests that the I term may be unnecessary in this application.

If the I (integrated error) term is dropped from the PID controller it becomes a purely feed-forward, stateless function, which would be very easy to implement.

The purpose of this notebook is to statistically characterise the PID controller function as a stateless, feed-forward function. This function could be compared to the simulated data to see what has been lost by dropping the I term.

It is worth emphasizing that the PID controller being modelled is completely fixed (i.e.it does not learn the control function and is not adaptive to any changes) and is not guaranteed to be optimally tuned.

1 Read data

This notebook is based on the data generated by the simulation runs with multiple target altitudes per run.

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 constrained to the range [x_min, x_max]
  {
  x %>% pmax(x_min) %>% pmin(x_max)
}

# function to extract a numeric parameter value from the file name
get_param_num <- 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()
}

# function to extract a sequence of numeric parameter values from the file name
get_param_num_seq <- 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 # character - character representation of a sequence, e.g. "(1,2,3)"
{
  file %>% str_match(regexp) %>% 
      subset(select = 2) %>% 
    str_replace_all(c("^" = "(", "_" = ",", "$" = ")")) # reformat as sequence
}

# function to extract a logical parameter value from the file name
get_param_log <- 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 *must* be T or F
) # value # logical - vector of logical parameter values
{
  file %>% str_match(regexp) %>% 
    subset(select = 2) %>% as.character() %>% "=="("T")
}

# read the data
d_wide <- fs::dir_ls(path = here::here("data", "multiple_target"), regexp = "/targets=.*\\.csv$") %>% # get file paths
  vroom::vroom(id = "file") %>% # read files
  dplyr::rename(k_tgt = target) %>% # rename for consistency with single target data
  dplyr::mutate( # add extra columns
    file = file %>% fs::path_ext_remove() %>% fs::path_file(), # get file name
    # get parameters
    targets  = file %>% get_param_num_seq("targets=([._0-9]+)_start="), # hacky
    k_start  = file %>% get_param_num("start=([.0-9]+)"), 
    sim_id   = paste(targets, k_start), # short ID for each simulation
    k_p      = file %>% get_param_num("kp=([.0-9]+)"),
    k_i      = file %>% get_param_num("Ki=([.0-9]+)"),
    # k_tgt    = file %>% get_param_num("k_tgt=([.0-9]+)"), # no longer needed
    k_windup = file %>% get_param_num("k_windup=([.0-9]+)"),
    uclip    = FALSE, # constant across all files
    dz0      = TRUE, # constant across all files
    # Deal with the fact that the interpretation of the imported u value
    # depends on the uclip parameter
    u_import = u, # keep a copy of the imported value to one side
    u = dplyr::if_else(uclip, # make u the "correct" value
                       u_import, 
                       clip(u_import, 0, 1)
                       ),
    # reconstruct the missing nodes
    i1 = k_tgt - z,
    i2 = i1 - dz,
    i3 = e * k_p,
    i9 = lag(ei, n = 1, default = 0), # initialised to zero
    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: 6000 Columns: 8
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (7): time, target, 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: 6,000
Columns: 27
$ file     <chr> "targets=1_3_5_start=3_kp=0.20_Ki=3.00_k_windup=0.20", "targe…
$ time     <dbl> 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0…
$ k_tgt    <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ z        <dbl> 3.000, 3.000, 3.003, 3.005, 3.006, 3.006, 3.005, 3.002, 2.999…
$ dz       <dbl> 0.000, 0.286, 0.188, 0.090, -0.008, -0.106, -0.204, -0.302, -…
$ e        <dbl> -2.000, -2.286, -2.191, -2.095, -1.998, -1.899, -1.800, -1.70…
$ ei       <dbl> -0.200, -0.200, -0.200, -0.200, -0.200, -0.200, -0.200, -0.20…
$ u        <dbl> 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000, 0.000…
$ targets  <chr> "(1,3,5)", "(1,3,5)", "(1,3,5)", "(1,3,5)", "(1,3,5)", "(1,3,…
$ k_start  <dbl> 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3…
$ sim_id   <chr> "(1,3,5) 3", "(1,3,5) 3", "(1,3,5) 3", "(1,3,5) 3", "(1,3,5) …
$ 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_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…
$ uclip    <lgl> FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE…
$ dz0      <lgl> TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, T…
$ u_import <dbl> -1.000, -1.057, -1.038, -1.019, -1.000, -0.980, -0.960, -0.94…
$ i1       <dbl> -2.000, -2.000, -2.003, -2.005, -2.006, -2.006, -2.005, -2.00…
$ i2       <dbl> -2.000, -2.286, -2.191, -2.095, -1.998, -1.900, -1.801, -1.70…
$ i3       <dbl> -0.4000, -0.4572, -0.4382, -0.4190, -0.3996, -0.3798, -0.3600…
$ i9       <dbl> 0.000, -0.200, -0.200, -0.200, -0.200, -0.200, -0.200, -0.200…
$ i4       <dbl> -2.000, -2.486, -2.391, -2.295, -2.198, -2.099, -2.000, -1.90…
$ i5       <dbl> -0.200, -0.200, -0.200, -0.200, -0.200, -0.200, -0.200, -0.20…
$ i6       <dbl> -0.600, -0.600, -0.600, -0.600, -0.600, -0.600, -0.600, -0.60…
$ i7       <dbl> -1.0000, -1.0572, -1.0382, -1.0190, -0.9996, -0.9798, -0.9600…
$ i8       <dbl> 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.000…
$ t        <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18…

2 Plot the data

The altitude error (\(i1\)) is a deterministic function of the altitude (\(z\)) and the altitude target (\(k_{tgt}\)). The two inputs \(z\) and \(k_{tgt}\) can be replaced with the single input \(i1\). This has the advantage of meaning that there are only two inputs \(i1\), and \(dz\), which can be represented on a plane.

Display the distribution of input values to the PID controller across all simulation runs.

d_wide %>% 
  ggplot(aes(x = i1, y = dz)) +
  geom_abline(slope = 1, intercept = 0, colour = "red") + # reference line
  geom_point(alpha = 0.1) +
  coord_equal()

Version Author Date
bb5f386 Ross Gayler 2021-09-09
e1e1d25 Ross Gayler 2021-08-20

  • The dark (because there are many overlapping points) diagonal line corresponds to the final phase of all the trajectories where the altitude error and vertical velocity both decrease to zero with any approximately constant ratio between the two quantities.

    • The ratio is 1:1. That is, the line corresponds to \(i2 (\triangleq i1n- dz) = 0\).
    • The central diagonal line deviates from \(i2 = 0\) for \(dz \lt -2.5\). This appears to be due to slight overshoot of the initial trajectories leading into the diagonal.

  • The light, approximately vertical curves correspond to the initial phase of all the trajectories, where the multicopter is either climbing at full power or dropping at zero power.

Rotate the axes by 45 degrees to make the diagonal axis-parallel in the new coordinates.

d_wide %>% 
  ggplot(aes( x = i1 - dz, y = (i1 + dz)/2)) +
  geom_point(alpha = 0.1) +
  coord_equal()

Version Author Date
bb5f386 Ross Gayler 2021-09-09
e1e1d25 Ross Gayler 2021-08-20
  • The x axis is equivalent to \(i2\).

Add the clipped motor command to the plot.

The clipped motor command lies in the range \([0, 1]\) and 0.524 is the command level for hovering. It will be useful to rescale the motor command level so that zero corresponds to hover and the magnitude of the maximum deviation from hover is one.

p <- d_wide %>% 
  dplyr::mutate(i8_scl = (i8 - 0.524) / 0.524) %>% 
  ggplot(aes(group = paste(sim_id, k_tgt))) +
  scale_color_viridis_c()

p +
  geom_point(aes(x = i1 - dz, y = (i1 + dz)/2, 
                 size = abs(i8_scl), colour = i8_scl),
             alpha = 0.5) +
  geom_path(aes(x = i1 - dz, y = (i1 + dz)/2), alpha = 0.2)

Version Author Date
bb5f386 Ross Gayler 2021-09-09
e1e1d25 Ross Gayler 2021-08-20

Remember, \(e \triangleq i2 \triangleq i1 - dz\). \(e\) is the “error”, which should be interpreted as the predicted altitude error in one second, assuming that the multicopter maintains the current vertical velocity.

This graph is a plan view, looking down on the surface to be modelled as a function of the input variables.

  • When \(e \lt 0\) (the left half-plane), the motor command \(i8\) is minimum.
  • When \(e \gt 0\) (the right half-plane), the motor command \(i8\) is (approximately) maximum.
  • When \(e \approx 0\) (the final phase of the trajectories) the motor command decreases as the \(y\) axis increases.
p +
  geom_point(aes(x = i1 - dz, y = i8_scl, 
                 size = abs(i8_scl), colour = i8_scl), 
             alpha = 0.5) +
  geom_path(aes(x = i1 - dz, y = i8_scl), alpha = 0.2)

Version Author Date
0b43b6e Ross Gayler 2021-09-09

This graph is a side view of the surface to be modelled. The \(x\) axis corresponds to \(e\) and the orthogonal input (\((i1 + dz)/2)\)) is collapsed over.

To a first approximation:

  • When \(e \lt 0\) (the left half-plane), the motor command \(i8\) is minimum.

  • When \(e \gt 0\) (the right half-plane), the motor command \(i8\) is (approximately) maximum.

    • The aproximation here is that as \(e\) approaches zero from above, when it gets sufficiently close to zero the motor command scales down from the maximum.
p +
  geom_abline(slope = -1/20, intercept = 0, colour = "red") + # reference line
  geom_point(aes(x = (i1 + dz)/2, y = i8_scl, 
                 size = abs(i8_scl), colour = i8_scl), 
             alpha = 0.5) +
  geom_path(aes(x = (i1 + dz)/2, y = i8_scl), alpha = 0.2)

Version Author Date
0b43b6e Ross Gayler 2021-09-09

This graph is the other side view of the surface to be modelled. The \(x\) axis corresponds to (\((i1 + dz)/2)\)) and the orthogonal input \(e\) is collapsed over.

  • Ignoring the overshoots and oscillations, when the trajectories are in the final phase (\(e \approx 0\)), the motor command is a linear function of \((i1 + dz)/2\).
  • The trajectories “drop” suddenly from the initial phase to the final phase.

2.1 Summary

  • Ignoring the overshoots and oscillations the overall shape of the function surface is not very complex (e.g. it’s not fractal).
  • The surface does involve an interaction. (It can’t be modelled as the sum of axis-parallel effects.)
  • There are sharp discontinuities.
  • The data from the initial phases is rather sparsely distributed over the input plane. This means that “learning” the function will require some strong biases to paper over the gaps.

3 Model the data

Fitting a smoothing spline regression to the data, it is very helpful if the predictor variables capture the major features of the data (to minimise the work done by the smoothing). Using an appropriately crafted set of predictor variables provides a bias to the functions that can be learned from the data.

I will use Generalised Additive Models (GAM) as the modelling technique here.

3.1 Without feature engineering

Show what a GAM can do without any problem-specific feature engineering.

Fit marginal smooths first to help build up intuition.

3.1.1 \(e\) term

Remember that \(u \triangleq i8\).

Model \(u\) as a function of \(e\).

fit1 <- mgcv::gam(u ~ s(e, bs =  "ad"), data = d_wide)
summary(fit1)

Family: gaussian 
Link function: identity 

Formula:
u ~ s(e, bs = "ad")

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.507238   0.000639   793.8   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Approximate significance of smooth terms:
      edf Ref.df    F p-value    
s(e) 18.9  21.57 1673  <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

R-sq.(adj) =  0.857   Deviance explained = 85.8%
GCV = 0.0024582  Scale est. = 0.0024501  n = 6000
plot(fit1)

Version Author Date
470fc77 Ross Gayler 2021-09-10

This uses an adaptive smoother (bs = "ad") so that the degrees of freedom are concentrated where the curvature is highest.

The scaling on the \(y\) axis of the plot is arbitrary.

  • The smooth picks up the strong dependency of motor command \(u\) on the sign of the error \(e\).
  • ~86% deviance explained. (Most of the variance of the motor command is explained by the \(e\) term.)

3.1.2 \(e_{orth}\) term

I will use \(e_{orth}\) as a convenience to refer to \((i1 + dz)/2\) because it is orthogonal to \(e\).

Model \(u\) as a function of \(e_{orth}\).

fit2 <- mgcv::gam(u ~ s(I((i1 + dz)/2), bs =  "ad"), data = d_wide)
summary(fit2)

Family: gaussian 
Link function: identity 

Formula:
u ~ s(I((i1 + dz)/2), bs = "ad")

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.507238   0.001555   326.1   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Approximate significance of smooth terms:
                    edf Ref.df     F p-value    
s(I((i1 + dz)/2)) 12.61  15.29 72.03  <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

R-sq.(adj) =  0.155   Deviance explained = 15.7%
GCV = 0.014548  Scale est. = 0.014515  n = 6000
plot(fit2)

Version Author Date
470fc77 Ross Gayler 2021-09-10
  • The central, slightly negatively sloped section from -2 to +2 captures the relation ship to \(u\) in the terminal phases of the trajectories (\(e \approx 0\)).
  • Outside that range, most of the observations are in the initial phases of the trajectories, so tend to be at extreme values of \(u\).
  • ~16% deviance explained. (\(e_{orth}\) only explains a small fraction of of the variance of the motor command, and even this might be dominated by the points outside the terminal phases of the trajectories.)

Fit the model on a subset of the observations corresponding to the terminal phases.

fit3 <- mgcv::gam(u ~ s(I((i1 + dz)/2), bs =  "cr"), data = d_wide, 
                  subset = abs(e) < 0.1)
plot(fit3)

Version Author Date
470fc77 Ross Gayler 2021-09-10
  • The negative slope in the terminal phases is clear. (The nonmonotonicity at the left end is due to a small number of outlier points).

3.1.3 Joint smooth

Fit a smooth that is a function of the joint values of \(e\) and \(e_{orth}\).

Use a thin-plate smoothing basis (bs = "tp").

fit4 <- mgcv::gam(u ~ s(e, I((i1 + dz)/2), bs =  "tp"), data = d_wide)
summary(fit4)

Family: gaussian 
Link function: identity 

Formula:
u ~ s(e, I((i1 + dz)/2), bs = "tp")

Parametric coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.5072383  0.0005842   868.3   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Approximate significance of smooth terms:
                      edf Ref.df    F p-value    
s(e,I((i1 + dz)/2)) 28.84     29 1529  <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

R-sq.(adj) =  0.881   Deviance explained = 88.1%
GCV = 0.0020577  Scale est. = 0.0020475  n = 6000
plot(fit4, se = FALSE, scheme = 2)

Version Author Date
470fc77 Ross Gayler 2021-09-10 
plot(fit4, se = FALSE, scheme = 1, theta = -30)

Version Author Date
470fc77 Ross Gayler 2021-09-10
  • The smooth captures the strong dependency between \(u\) and \(e\).
  • The smooth fails to capture the relationship with \(e_{orth}\) at \(e \approx 0\).
  • ~88% deviance explained. (This is only a slight increase relative to modelling only the \(e\) term.)

3.2 With feature engineering

Model the function, this time using features designed to capture the major aspects of the shape of the surface. Such features constitute a strong bias on the set of functions that can be learned from the data. They also provide a mechanism for extrapolating outside the support of the data.

3.2.1 \(e\) terms

The previous analyses showed that the surface should be partitioned into three regions along \(e\).

  • When \(e < 0\) then \(u\) is constant at 0 (initial phase of trajectories).
  • When \(e > 0\) then \(u\) is roughly constant at (initial phase of trajectories).
  • When \(e = 0\) then \(u\) is roughly a linear function of \(e_{orth}\) (terminal phase of trajectories).

The \(e = 0\) class is infinitesimally narrow, which is likely to cause problems in a practical system. Consequently, I will make that class a small finite width.

e_fuzz <- 0.1 # half width of central strip
fit5 <- mgcv::gam(u ~ I(e >= e_fuzz) + I(abs(e) < e_fuzz), data = d_wide)
summary(fit5)

Family: gaussian 
Link function: identity 

Formula:
u ~ I(e >= e_fuzz) + I(abs(e) < e_fuzz)

Parametric coefficients:
                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)            6.661e-16  3.917e-03     0.0        1    
I(e >= e_fuzz)TRUE     7.351e-01  5.177e-03   142.0   <2e-16 ***
I(abs(e) < e_fuzz)TRUE 5.164e-01  4.005e-03   128.9   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


R-sq.(adj) =  0.781   Deviance explained = 78.1%
GCV = 0.0037613  Scale est. = 0.0037594  n = 6000

Scoring out regions by eye from the regression coefficients (this is a step function):

  • \(e \le -0.1\) - \(u \approx 0\)

  • \(-0.1 \lt e \lt +0.1\) - \(u \approx 0.52\)

  • \(e \ge +0.1\) - \(u \approx 0.74\)

  • ~78% deviance explained (compared to 86% for the smoothing spline model).

Look at the plot of fit1. The smoothing spline model has a slope between $e and \(e \approx 2\). The current model (fit5) can’t account for this slope, which is why the goodness of fit is reduced.

However, it’s not clear that the slope is actually needed for the controller to work well, so I’ll ignore it.

3.2.2 \(e_{orth}\) term

Now add the \(e_{orth}\) into the model.

\(u\) is a linear function of \(e_{orth}\) when \(e \approx 0\) and irrelevant elsewhere, which requires an interaction of \(e_{orth}\) with \(e\).

fit6 <- mgcv::gam(
  u ~ I(e >= e_fuzz) + I(abs(e) < e_fuzz) +
    I(as.numeric(abs(e) < e_fuzz) * (i1 + dz)/2), 
  data = d_wide)
summary(fit6)

Family: gaussian 
Link function: identity 

Formula:
u ~ I(e >= e_fuzz) + I(abs(e) < e_fuzz) + I(as.numeric(abs(e) < 
    e_fuzz) * (i1 + dz)/2)

Parametric coefficients:
                                               Estimate Std. Error t value
(Intercept)                                  -3.886e-16  3.433e-03    0.00
I(e >= e_fuzz)TRUE                            7.351e-01  4.537e-03  162.02
I(abs(e) < e_fuzz)TRUE                        5.214e-01  3.511e-03  148.50
I(as.numeric(abs(e) < e_fuzz) * (i1 + dz)/2) -2.665e-02  6.258e-04  -42.59
                                             Pr(>|t|)    
(Intercept)                                         1    
I(e >= e_fuzz)TRUE                             <2e-16 ***
I(abs(e) < e_fuzz)TRUE                         <2e-16 ***
I(as.numeric(abs(e) < e_fuzz) * (i1 + dz)/2)   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


R-sq.(adj) =  0.832   Deviance explained = 83.2%
GCV = 0.0028886  Scale est. = 0.0028866  n = 6000

Scoring out regions by eye from the regression coefficients (this is a step function):

  • \(e \le -0.1\) - \(u \approx 0\)

  • \(-0.1 \lt e \lt +0.1\) - \(u \approx -0.027 e_{orth} + 0.52\)

  • \(e \ge +0.1\) - \(u \approx 0.74\)

  • ~83% deviance explained (compared to 88% for the bivariate smoothing spline model).

This model seems to pick up the major features of the surface.

In a future analysis I will use the construction of these terms to guide the placement of the knots in VSA representations of scalars.


sessionInfo()
R version 4.1.1 (2021-08-10)
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] tibble_3.1.4       Matrix_1.3-4       purrr_0.3.4        DiagrammeR_1.0.6.1
 [5] mgcv_1.8-36        nlme_3.1-153       ggplot2_3.3.5      stringr_1.4.0     
 [9] dplyr_1.0.7        vroom_1.5.4        here_1.0.1         fs_1.5.0          

loaded via a namespace (and not attached):
 [1] tidyselect_1.1.1   xfun_0.25          splines_4.1.1      lattice_0.20-44   
 [5] colorspace_2.0-2   vctrs_0.3.8        generics_0.1.0     viridisLite_0.4.0 
 [9] htmltools_0.5.2    yaml_2.2.1         utf8_1.2.2         rlang_0.4.11      
[13] later_1.3.0        pillar_1.6.2       glue_1.4.2         withr_2.4.2       
[17] bit64_4.0.5        RColorBrewer_1.1-2 lifecycle_1.0.0    munsell_0.5.0     
[21] gtable_0.3.0       workflowr_1.6.2    visNetwork_2.0.9   htmlwidgets_1.5.4 
[25] evaluate_0.14      labeling_0.4.2     knitr_1.34         tzdb_0.1.2        
[29] fastmap_1.1.0      httpuv_1.6.3       parallel_4.1.1     fansi_0.5.0       
[33] highr_0.9          Rcpp_1.0.7         renv_0.14.0        promises_1.2.0.1  
[37] scales_1.1.1       jsonlite_1.7.2     farver_2.1.0       bit_4.0.4         
[41] digest_0.6.27      stringi_1.7.4      bookdown_0.24      rprojroot_2.0.2   
[45] grid_4.1.1         cli_3.0.1          tools_4.1.1        magrittr_2.0.1    
[49] crayon_1.4.1       whisker_0.4        pkgconfig_2.0.3    ellipsis_0.3.2    
[53] rstudioapi_0.13    rmarkdown_2.10     R6_2.5.1           git2r_0.28.0      
[57] compiler_4.1.1