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trajectory calculator
trajectory_pars <- list(
# constants
mass = 5.125, # oz,
circumference = 9.125, # in
beta = 1.217e-4, # 1 / meter
cd0 = 0.3008,
cdspin = 0.0292,
cl0 = 0.583,
cl1 = 2.333,
cl2 = 1.120,
tau = 10000, # seconds
g_gravity = 32.174,
# conversions,
mph_to_fts = 1.467,
ft_to_m = 0.3048037,
lbft3_to_kgm3 = 16.01848,
kgm3_to_lbft3 = 0.06242789,
# environmental parameters
vwind = 0, # mph
phiwind = 0, #deg
hwind = 0, #ft
relative_humidity = 50,
pressure_in_hg = 29.92,
temperature_f = 70, #F
elevation_ft= 15, #feet,
# batted ball parameters
x0 = 0, # ft
y0 = 2.0, # ft
z0 = 3.0, # ft
spin = 2675, # revs per second
spin_phi = -18.5, # degrees
drag_strength = 1, # 1 = full, 0 to disable
magnus_strength = 1,
tmp = 1
)
compute_pars <- function(pars) {
pars$elevation_m = pars$elevation_ft * pars$ft_to_m
pars$temperature_c = (pars$temperature_f - 32) * 5 / 9
pars$pressure_mm_hg = pars$pressure_in_hg * 1000/39.37
pars$RH = pars$relative_humidity
pars$SVP = 4.5841*exp((18.687-pars$temperature_c/234.5)*pars$temperature_c/(257.14+pars$temperature_c))
pars$rho = 1.2929 * (273/(pars$temperature_c+273) *
(pars$pressure_mm_hg*exp(-pars$beta * pars$elevation_m) - 0.3783*pars$RH*pars$SVP * 0.01) / 760)
pars$c0 = 0.07182 * pars$rho * pars$kgm3_to_lbft3 * (5.125/pars$mass)*(pars$circumference/9.125)**2
pars$sidespin = pars$spin * sin(pars$spin_phi * pi/180)
pars$backspin = pars$spin * cos(pars$spin_phi * pi/180)
pars$omega = pars$spin * pi / 30
pars$romega = pars$circumference * pars$omega / (24 * pi)
pars
}
rk2 <- function(f, xn, tn, h) {
k1 = h * f(tn, xn)
k2 = h * f(tn + h/2, xn + k1/2)
xn_1 = xn + k2
}
rk4 <- function(f, xn, tn, h) {
k1 = h*f(tn,xn)
k2 = h*f(tn + h/2, xn + k1/2)
k3 = h*f(tn + h/2, xn + k2/2)
k4 = h*f(tn + h, xn + k3)
xn_1 = xn + k1/6 + k2/3 + k3/3 + k4/6
}
s_fun <- function(t, vw, pars) {
(pars$romega / vw) * exp(-t *vw /(pars$tau*146.7))
}
cl_fun <- function(t, vw, pars) {
s = s_fun(t, vw, pars)
pars$cl2*s/(pars$cl0+pars$cl1*s)
}
cd_fun <- function(t, vw, pars) {
pars$cd0 + pars$cdspin * (pars$spin * 1e-3)*exp(-t * vw/(pars$tau*146.7))
}
projectile1 = function(v_initial, launch_angle, launch_phi, dt, pars, N=1e4, stop_dim=3, ...) {
pars$launch_angle = launch_angle
pars$launch_phi = launch_phi
pars = compute_pars(pars)
extra_pars = list(...)
for (extra_par in names(extra_pars)) {
pars[[extra_par]] = extra_pars[[extra_par]]
}
t0 = 0.0
phi = pars$launch_phi
theta = pars$launch_angle
phi_rad = pars$launch_phi * pi / 180
theta_rad = pars$launch_angle * pi / 180
xc = c(pars$x0, pars$y0, pars$z0)
vc = v_initial * pars$mph_to_fts * c(cos(theta_rad) * sin(phi_rad), cos(theta_rad) * cos(phi_rad), sin(theta_rad))
init_stop = as.integer(xc[[stop_dim]] > 0)
now_stop = as.integer(xc[[stop_dim]] > 0)
xa = matrix(rep(0, 3 * N), ncol=3)
va = matrix(rep(0, 3 * N), ncol=3)
fa = matrix(rep(0, 9 * N), ncol=9)
ta = matrix(rep(0, 1 * N), ncol=1)
cda = matrix(rep(0, 3 * N), ncol=3)
wa = matrix(rep(0, 3 * N), ncol=3)
i = 0
while(now_stop == init_stop && i < N) {
if (i %% 1000 == 0) {
message(i)
}
i = i + 1
tc = t0 + i*dt
ta[i,1] = tc
vx = vc[[1]]
vy = vc[[2]]
vz = vc[[3]]
v = sqrt(vx**2 + vy**2 + vz**2)
wb = pars$backspin
ws = pars$sidespin
wx = (wb * cos(phi * pi/180) - ws * sin(theta * pi / 180) * sin(phi * pi/180)) * pi / 30
wy = (-wb * sin(phi * pi/180) - ws * sin(theta * pi / 180) * cos(phi * pi/180)) * pi / 30
wz = (ws * cos(theta * pi/180)) * pi / 30
wa[i, 1] = wx
wa[i, 2] = wy
wa[i, 3] = wz
cd = cd_fun(tc, v, pars)
cl = cl_fun(tc, v, pars)
s = s_fun(tc, v, pars)
magnus_const = pars$c0 * cl/pars$omega * v
magnus_const = magnus_const * pars$magnus_strength
cda[i, 1] = cd
cda[i, 2] = cl
cda[i, 3] = s
drag_const = pars$c0 * cd * v
drag_const = pars$drag_strength * drag_const
fx = function(t, x) {
-drag_const * vx + magnus_const * (wy * vz - wz * vy)
}
fy = function(t, x) {
-drag_const * vy + magnus_const * (-wx * vz + wz * vx)
}
fz = function(t, x) {
-drag_const * vz + magnus_const * (wx * vy - wy * vx) - pars$g_gravity
}
gx = function(a, b) {vx}
gy = function(a, b) {vy}
gz = function(a, b) {vz}
vc[[1]] = rk4(fx, vc[[1]], tc, dt)
vc[[2]] = rk4(fy, vc[[2]], tc, dt)
vc[[3]] = rk4(fz, vc[[3]], tc, dt)
xc[[1]] = rk4(gx, xc[[1]], tc, dt)
xc[[2]] = rk4(gy, xc[[2]], tc, dt)
xc[[3]] = rk4(gz, xc[[3]], tc, dt)
# message(tc, " ", xc[[j]], " ", vc[[j]])
xa[i, 1:3] = xc
va[i, 1:3] = vc
fa[i, 1:9] = c(fx(tc, vc[[1]]), fy(tc, vc[[2]]), fz(tc, vc[[3]]),
-drag_const * vx,
-drag_const * vy,
-drag_const * vz,
magnus_const * (wy * vz - vy * wz),
magnus_const * (vx * wz - vz * wx),
magnus_const * (wx * vy - wy * vx)
)
now_stop = as.integer(xc[[stop_dim]] > 0)
}
data.frame(t=ta[1:i,1],
x=xa[1:i, 1], y=xa[1:i, 2], z=xa[1:i, 3],
vx=va[1:i, 1], vy=va[1:i, 2], vz=va[1:i, 3],
ax=fa[1:i, 1], ay=fa[1:i, 2], az=fa[1:i, 3],
adragx=fa[1:i, 4], adragy=fa[1:i, 5], adragz=fa[1:i, 6],
aMagx=fa[1:i, 7], aMagy=fa[1:i, 8], aMagz=fa[1:i, 9],
Cd=cda[1:i, 1], Cl=cda[1:i, 2], S=cda[1:i, 3])
}
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