/
optimal-functions.R
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optimal-functions.R
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#' Optimal profile functions
#'
#' Family of functions that serve a purpose of finding optimal sprint or force-velocity profile
#'
#' @param MSS,MAC Numeric vectors. Model parameters
#' @param F0,V0 Numeric vectors. FV profile parameters
#' @param bodymass Body mass in kg
#' @param inertia External inertia in kg (for example a weight vest, or a sled).
#' Not included in the air resistance calculation
#' @param resistance External horizontal resistance in Newtons (for example tether device or a sled friction resistance)
#' @param distance Numeric vector
#' @inheritDotParams get_air_resistance
#' @references
#' Samozino P, Peyrot N, Edouard P, Nagahara R, Jimenez‐Reyes P, Vanwanseele B, Morin J. 2022.
#' Optimal mechanical force-velocity profile for sprint acceleration performance.
#' Scandinavian Journal of Medicine & Science in Sports 32:559–575. DOI: 10.1111/sms.14097.
#' @examples
#' MSS <- 10
#' MAC <- 8
#' bodymass <- 75
#'
#' fv <- create_FVP(MSS, MAC, bodymass)
#'
#' dist <- seq(5, 40, by = 5)
#'
#' opt_MSS_MAC_profile <- optimal_MSS_MAC(
#' distance = dist,
#' MSS,
#' MAC
#' )[["profile_imb"]]
#'
#' opt_FV_profile <- optimal_FV(
#' distance = dist,
#' fv$F0,
#' fv$V0,
#' fv$bodymass
#' )[["profile_imb"]]
#'
#' opt_FV_profile_peak <- optimal_FV(
#' distance = dist,
#' fv$F0,
#' fv$V0,
#' fv$bodymass,
#' method = "peak"
#' )[["profile_imb"]]
#'
#' plot(x = dist, y = opt_MSS_MAC_profile, type = "l", ylab = "Profile imbalance")
#' lines(x = dist, y = opt_FV_profile, type = "l", col = "blue")
#' lines(x = dist, y = opt_FV_profile_peak, type = "l", col = "red")
#' abline(h = 100, col = "gray", lty = 2)
#' @name optimal_functions
NULL
# ---------------------------------
# Functions to find optimal FV profile
# Scalar functions
# Function to find optimal profile while keeping the arithmetic Max Power the same
find_optimal_FV_scalar <- function(distance, F0, V0, bodymass = 75, inertia = 0, resistance = 0, ...) {
opt_func <- function(par) {
new_F0 <- F0 / par[1]
new_V0 <- V0 * par[1]
predict_time_at_distance_FV(
distance = distance,
F0 = new_F0,
V0 = new_V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
}
get_optim_model <- function() {
tryCatch(
{
stats::optim(
par = 1,
fn = opt_func,
method = "Brent",
lower = 1 / min(100, F0 / ifelse(resistance < 0, 0, resistance)),
upper = min(100, F0 / ifelse(resistance < 0, 0, resistance))
)
},
error = function(cond) {
return(list(par = NA, value = NA))
},
warning = function(cond) {
return(list(par = NA, value = NA))
}
)
}
results <- get_optim_model()
F0_optim <- F0 / results$par
V0_optim <- V0 * results$par
FV_slope <- -(F0 / bodymass) / V0
FV_slope_optim <- -(F0_optim / bodymass) / V0_optim
profile_imb <- (FV_slope / FV_slope_optim) * 100
t_orig <- predict_time_at_distance_FV(
distance = distance,
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
##############################################
# Peak Power
# Original profile
converted <- convert_FVP(
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
MSS_conv <- converted$MSS
MAC_conv <- converted$MAC
Ppeak_dist <- find_peak_power_distance(
MSS = MSS_conv,
MAC = MAC_conv,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
Ppeak_time <- find_peak_power_time(
MSS = MSS_conv,
MAC = MAC_conv,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
# Optimal profile
converted_optim <- convert_FVP(
F0 = F0_optim,
V0 = V0_optim,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
MSS_conv_optim <- converted_optim$MSS
MAC_conv_optim <- converted_optim$MAC
Ppeak_dist_optim <- find_peak_power_distance(
MSS = MSS_conv_optim,
MAC = MAC_conv_optim,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
Ppeak_time_optim <- find_peak_power_time(
MSS = MSS_conv_optim,
MAC = MAC_conv_optim,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
# Return the results
list(
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
Pmax = F0 * V0 / 4,
Pmax_rel = (F0 * V0 / 4) / bodymass,
slope = FV_slope,
distance = distance,
time = t_orig,
Ppeak = Ppeak_dist$peak_power,
Ppeak_rel = Ppeak_dist$peak_power / bodymass,
Ppeak_dist = Ppeak_dist$distance,
Ppeak_time = Ppeak_time$time,
F0_optim = F0_optim,
F0_coef = 1 / results$par,
V0_optim = V0_optim,
V0_coef = results$par,
Pmax_optim = F0_optim * V0_optim / 4,
Pmax_rel_optim = (F0_optim * V0_optim / 4) / bodymass,
slope_optim = FV_slope_optim,
profile_imb = profile_imb,
time_optim = results$value,
time_gain = results$value - t_orig,
Ppeak_optim = Ppeak_dist_optim$peak_power,
Ppeak_rel_optim = Ppeak_dist_optim$peak_power / bodymass,
Ppeak_dist_optim = Ppeak_dist_optim$distance,
Ppeak_time_optim = Ppeak_time_optim$time
)
}
# Assistant function for find_optimal_FV_peak_scalar()
find_FV_peak_power <- function(F0, V0, bodymass = 75, inertia = 0, resistance = 0, ...) {
converted <- convert_FVP(
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
MSS_conv <- converted$MSS
MAC_conv <- converted$MAC
find_peak_power_distance(
MSS = MSS_conv,
MAC = MAC_conv,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)$peak_power
}
# Assistant function for find_optimal_FV_peak_scalar()
find_V0 <- function(F0, Ppeak, bodymass = 75, inertia = 0, resistance = 0, ...) {
opt_func <- function(V0) {
(Ppeak - find_FV_peak_power(
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
))
}
V0 <- (4 * Ppeak) / F0
get_optim_model <- function() {
tryCatch(
{
stats::uniroot(f = opt_func, interval = c(1 / 2, 2) * V0)$root
},
error = function(cond) {
return(NA)
},
warning = function(cond) {
return(NA)
}
)
}
get_optim_model()
}
# Function to find optimal profile while keeping the manifested Peak Power the same
find_optimal_FV_peak_scalar <- function(distance, F0, V0, bodymass = 75, inertia = 0, resistance = 0, ...) {
Ppeak_orig <- find_FV_peak_power(
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
opt_func <- function(par) {
new_F0 <- par
new_V0 <- find_V0(
F0 = new_F0,
Ppeak = Ppeak_orig,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
predict_time_at_distance_FV(
distance = distance,
F0 = new_F0,
V0 = new_V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
}
F0_optim <- find_optimal_FV_scalar(
distance = distance,
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)$F0_optim
get_optim_model <- function() {
tryCatch(
{
stats::optim(
par = F0_optim,
fn = opt_func,
method = "Brent",
lower = F0_optim * 0.8,
upper = F0_optim * 1.2
)
},
error = function(cond) {
return(list(par = NA, value = NA))
},
warning = function(cond) {
return(list(par = NA, value = NA))
}
)
}
results <- get_optim_model()
F0_optim <- results$par
V0_optim <- find_V0(
F0 = F0_optim,
Ppeak = Ppeak_orig,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
FV_slope <- -(F0 / bodymass) / V0
FV_slope_optim <- -(F0_optim / bodymass) / V0_optim
profile_imb <- (FV_slope / FV_slope_optim) * 100
# Probe
t_orig <- predict_time_at_distance_FV(
distance = distance,
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
##############################################
# Peak Power
# Original profile
converted <- convert_FVP(
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
MSS_conv <- converted$MSS
MAC_conv <- converted$MAC
Ppeak_dist <- find_peak_power_distance(
MSS = MSS_conv,
MAC = MAC_conv,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
Ppeak_time <- find_peak_power_time(
MSS = MSS_conv,
MAC = MAC_conv,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
# Optimal profile
converted_optim <- convert_FVP(
F0 = F0_optim,
V0 = V0_optim,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
MSS_conv_optim <- converted_optim$MSS
MAC_conv_optim <- converted_optim$MAC
Ppeak_dist_optim <- find_peak_power_distance(
MSS = MSS_conv_optim,
MAC = MAC_conv_optim,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
Ppeak_time_optim <- find_peak_power_time(
MSS = MSS_conv_optim,
MAC = MAC_conv_optim,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
...
)
# Return the results
list(
F0 = F0,
V0 = V0,
bodymass = bodymass,
inertia = inertia,
resistance = resistance,
Pmax = F0 * V0 / 4,
Pmax_rel = (F0 * V0 / 4) / bodymass,
slope = FV_slope,
distance = distance,
time = t_orig,
Ppeak = Ppeak_dist$peak_power,
Ppeak_rel = Ppeak_dist$peak_power / bodymass,
Ppeak_dist = Ppeak_dist$distance,
Ppeak_time = Ppeak_time$time,
F0_optim = F0_optim,
F0_coef = F0_optim / F0,
V0_optim = V0_optim,
V0_coef = V0_optim / V0,
Pmax_optim = F0_optim * V0_optim / 4,
Pmax_rel_optim = (F0_optim * V0_optim / 4) / bodymass,
slope_optim = FV_slope_optim,
profile_imb = profile_imb,
time_optim = results$value,
time_gain = results$value - t_orig,
Ppeak_optim = Ppeak_dist_optim$peak_power,
Ppeak_rel_optim = Ppeak_dist_optim$peak_power / bodymass,
Ppeak_dist_optim = Ppeak_dist_optim$distance,
Ppeak_time_optim = Ppeak_time_optim$time
)
}
# Vector function
#' @rdname optimal_functions
#' @description \code{optimal_FV} finds "optimal" \code{F0} and \code{V0} where time at distance is
#' minimized, while keeping the power the same
#' @param method Method to be utilized. Options are "peak" and "max" (default)
#' @return \code{optimal_FV} returns s data frame with the following columns
#' \describe{
#' \item{F0}{Original F0}
#' \item{V0}{Original F0}
#' \item{bodymass}{Bodymass}
#' \item{inertia}{Inertia}
#' \item{resistance}{Resistance}
#' \item{Pmax}{Maximal power estimated using F0 * V0 / 4}
#' \item{Pmax_rel}{Relative maximal power}
#' \item{slope}{FV profile slope}
#' \item{distance}{Distance}
#' \item{time}{Time to cover distance}
#' \item{Ppeak}{Peak power estimated quantitatively}
#' \item{Ppeak_rel}{Relative peak power}
#' \item{Ppeak_dist}{Distance at which peak power is manifested}
#' \item{Ppeak_time}{Time at which peak power is manifested}
#' \item{F0_optim}{Optimal F0}
#' \item{F0_coef}{Ratio between F0_optim an F0}
#' \item{V0_optim}{Optimal V0}
#' \item{V0_coef}{Ratio between V0_optim an V0}
#' \item{Pmax_optim}{Optimal maximal power estimated F0_optim * V0_optim / 4}
#' \item{Pmax_rel_optim}{Optimal relative maximal power}
#' \item{slope_optim}{Optimal FV profile slope}
#' \item{profile_imb}{Percent ratio between slope and optimal slope}
#' \item{time_optim}{Time to cover distance when profile is optimal}
#' \item{time_gain}{Difference in time to cover distance between time_optimal and time}
#' \item{Ppeak_optim}{Optimal peak power estimated quantitatively}
#' \item{Ppeak_rel_optim}{Optimal relative peak power}
#' \item{Ppeak_dist_optim}{Distance at which optimal peak power is manifested}
#' \item{Ppeak_time_optim}{Time at which optimal peak power is manifested}
#' }
#' @export
optimal_FV <- function(distance, F0, V0, bodymass = 75, inertia = 0, resistance = 0, method = "max", ...) {
df <- data.frame(
distance = distance,
F0 = F0,
V0 = V0,
bodymass = bodymass,
method = method,
inertia = inertia,
resistance = resistance,
...
)
df$id <- sprintf(paste0("%0", floor(log10(nrow(df))) + 1, "d"), seq(1, nrow(df)))
df_list <- split(df, df$id)
purrr::map_df(df_list, function(.x) {
if (!(.x$method %in% c("peak", "max"))) {
stop("Unknown method of the profile optimization. Please use 'max' or 'peak'")
}
if (.x$method == "max") {
optim_func <- find_optimal_FV_scalar
} else if (.x$method == "peak") {
optim_func <- find_optimal_FV_peak_scalar
}
.x$method <- NULL
.x$id <- NULL
res <- do.call(optim_func, as.list(.x))
data.frame(res)
})
}
# ------------------------------------------------
# Functions for optimal sprint profile
find_optimal_MSS_MAC_scalar <- function(distance, MSS, MAC) {
opt_func <- function(par) {
new_MSS <- MSS / par[1]
new_MAC <- MAC * par[1]
predict_time_at_distance(
distance = distance,
MSS = new_MSS,
MAC = new_MAC
)
}
get_optim_model <- function() {
tryCatch(
{
stats::optim(
par = 1,
fn = opt_func,
method = "Brent",
lower = 0,
upper = 10
)
},
error = function(cond) {
return(list(par = NA, value = NA))
},
warning = function(cond) {
return(list(par = NA, value = NA))
}
)
}
results <- get_optim_model()
MSS_optim <- MSS / results$par
MAC_optim <- MAC * results$par
slope <- -MAC / MSS
slope_optim <- -MAC_optim / MSS_optim
profile_imb <- (slope / slope_optim) * 100
# Probe
t_orig <- predict_time_at_distance(
distance = distance,
MSS = MSS,
MAC = MAC
)
# Return the results
list(
MSS = MSS,
MAC = MAC,
Pmax_rel = MSS * MAC / 4,
slope = slope,
distance = distance,
time = t_orig,
MSS_optim = MSS_optim,
MSS_coef = 1 / results$par,
MAC_optim = MAC_optim,
MAC_coef = results$par,
Pmax_rel_optim = MSS * MAC / 4,
slope_optim = slope_optim,
profile_imb = profile_imb,
time_optim = results$value,
time_gain = results$value - t_orig
)
}
#' @rdname optimal_functions
#' @description \code{optimal_MSS_MAC} finds "optimal" \code{MSS} and \code{MAS} where time at distance is
#' minimized, while keeping the \code{Pmax} the same
#' @return \code{optimal_MSS_MAC} returns a data frame with the following columns
#' \describe{
#' \item{MSS}{Original MSS}
#' \item{MAC}{Original MAC}
#' \item{Pmax_rel}{Relative maximal power estimated using MSS * MAC / 4}
#' \item{slope}{Sprint profile slope}
#' \item{distance}{Distance}
#' \item{time}{Time to cover distance}
#' \item{MSS_optim}{Optimal MSS}
#' \item{MSS_coef}{Ratio between MSS_optim an MSS}
#' \item{MAC_optim}{Optimal MAC}
#' \item{MAC_coef}{Ratio between MAC_optim an MAC}
#' \item{Pmax_rel_optim}{Optimal relative maximal power estimated using MSS_optim * MAC_optim / 4}
#' \item{slope_optim}{Optimal sprint profile slope}
#' \item{profile_imb}{Percent ratio between slope and optimal slope}
#' \item{time_optim}{Time to cover distance when profile is optimal}
#' \item{time_gain}{Difference in time to cover distance between time_optimal and time}
#' }
#' @export
optimal_MSS_MAC <- function(distance, MSS, MAC) {
df <- data.frame(
distance = distance,
MSS = MSS,
MAC = MAC
)
df$id <- sprintf(paste0("%0", floor(log10(nrow(df))) + 1, "d"), seq(1, nrow(df)))
df_list <- split(df, df$id)
purrr::map_df(df_list, function(.x) {
data.frame(find_optimal_MSS_MAC_scalar(
distance = .x$distance,
MSS = .x$MSS,
MAC = .x$MAC
))
})
}