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gotelem/py/pytelem/optimus.py

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# hyperspeed forward and backwards analytics engine
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import numpy as np
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from numba import jit
# import jax.numpy as np
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# TODO: define 3d vector space - x,y,z oriented around car/world?
# solvers should not care about position but we should be able to convert
# transforms between car/world spaces.
# let's define x as the forward-backward axis (with forward being positive)
# y as the lateral axis, with right being positive
# and z being the vertical, with up being positive.
# for simplicities’ sake, the car only rotates on the y-axis (up and down hills).
# z rotation can be determined from distance along the route.
# some data (wind) is only a 2d vector at a given time point.
def fsolve_discrete():
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"""Forward compute a route segment."""
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def dist_to_pos(dist: float):
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"""convert a distance along the race path to a position in 3d space"""
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# All units are BASE SI (no prefix except for kilogram)
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ATM_MOLAR_MASS = 0.0289644 # kg/mol
STANDARD_TEMP = 288.15 # K
STANDARD_PRES = 101325.0 # Pa
AIR_GAS_CONSTANT = 8.31432 # N*m/s^2
EARTH_TEMP_LAPSE = -0.0065
EARTH_GRAVITY = 9.80665 # m/s^2
EARTH_RADIUS = 6378140.0 # m
EARTH_AXIS_INCLINATION = 23.45 # degrees
# FIXME: use named constants here
@jit
def get_pressure_el(
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el: float,
Ps=STANDARD_PRES,
Ts: float = STANDARD_TEMP,
T_lapse: float = EARTH_TEMP_LAPSE,
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):
"""Gets the pressure at a point given eleveation - assumes
standard pressure,temperature, gas constants, etc"""
return Ps * (Ts / (Ts + T_lapse * el)) ** (
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(ATM_MOLAR_MASS * EARTH_GRAVITY) / (AIR_GAS_CONSTANT / T_lapse)
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)
@jit
def estimate_temp(el: float, Ts: float = STANDARD_TEMP, T_lapse=EARTH_TEMP_LAPSE):
return Ts + el * T_lapse
def make_cubic(a, b, c, d):
"""returns a simple cubic function"""
def poly(x):
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return a + b * x + c * (x ** 2) + (x ** 3) / d
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return jit(poly)
@jit
def get_radiation_direct(yday, altitude_deg):
"""Calculate the direct radiation at a given day of the year given the angle of the sun
from the horizon."""
flux = 1160 + (75 * np.sin(2 * np.pi / 365 * (yday - 275)))
optical_depth = 0.174 + (0.035 * np.sin(2 * np.pi / 365 * (yday - 100)))
air_mass_ratio = 1 / np.sin(np.radians(altitude_deg))
# from Masters, p. 412
return flux * np.exp(-1 * optical_depth * air_mass_ratio) * (altitude_deg > 0)
# We start by defining MANY constants.
# to skip this, Ctrl-F to END COEFF
# START COEFF
# heliocentric longitude, latitude, radius (section 3.2) coefficients
# heliocentric longitude coefficients
L0_TABLE = np.array(
[
[175347046.0, 0.0, 0.0],
[3341656.0, 4.6692568, 6283.07585],
[34894.0, 4.6261, 12566.1517],
[3497.0, 2.7441, 5753.3849],
[3418.0, 2.8289, 3.5231],
[3136.0, 3.6277, 77713.7715],
[2676.0, 4.4181, 7860.4194],
[2343.0, 6.1352, 3930.2097],
[1324.0, 0.7425, 11506.7698],
[1273.0, 2.0371, 529.691],
[1199.0, 1.1096, 1577.3435],
[990.0, 5.233, 5884.927],
[902.0, 2.045, 26.298],
[857.0, 3.508, 398.149],
[780.0, 1.179, 5223.694],
[753.0, 2.533, 5507.553],
[505.0, 4.583, 18849.228],
[492.0, 4.205, 775.523],
[357.0, 2.92, 0.067],
[317.0, 5.849, 11790.629],
[284.0, 1.899, 796.298],
[271.0, 0.315, 10977.079],
[243.0, 0.345, 5486.778],
[206.0, 4.806, 2544.314],
[205.0, 1.869, 5573.143],
[202.0, 2.458, 6069.777],
[156.0, 0.833, 213.299],
[132.0, 3.411, 2942.463],
[126.0, 1.083, 20.775],
[115.0, 0.645, 0.98],
[103.0, 0.636, 4694.003],
[102.0, 0.976, 15720.839],
[102.0, 4.267, 7.114],
[99.0, 6.21, 2146.17],
[98.0, 0.68, 155.42],
[86.0, 5.98, 161000.69],
[85.0, 1.3, 6275.96],
[85.0, 3.67, 71430.7],
[80.0, 1.81, 17260.15],
[79.0, 3.04, 12036.46],
[75.0, 1.76, 5088.63],
[74.0, 3.5, 3154.69],
[74.0, 4.68, 801.82],
[70.0, 0.83, 9437.76],
[62.0, 3.98, 8827.39],
[61.0, 1.82, 7084.9],
[57.0, 2.78, 6286.6],
[56.0, 4.39, 14143.5],
[56.0, 3.47, 6279.55],
[52.0, 0.19, 12139.55],
[52.0, 1.33, 1748.02],
[51.0, 0.28, 5856.48],
[49.0, 0.49, 1194.45],
[41.0, 5.37, 8429.24],
[41.0, 2.4, 19651.05],
[39.0, 6.17, 10447.39],
[37.0, 6.04, 10213.29],
[37.0, 2.57, 1059.38],
[36.0, 1.71, 2352.87],
[36.0, 1.78, 6812.77],
[33.0, 0.59, 17789.85],
[30.0, 0.44, 83996.85],
[30.0, 2.74, 1349.87],
[25.0, 3.16, 4690.48],
]
)
L1_TABLE = np.array(
[
[628331966747.0, 0.0, 0.0],
[206059.0, 2.678235, 6283.07585],
[4303.0, 2.6351, 12566.1517],
[425.0, 1.59, 3.523],
[119.0, 5.796, 26.298],
[109.0, 2.966, 1577.344],
[93.0, 2.59, 18849.23],
[72.0, 1.14, 529.69],
[68.0, 1.87, 398.15],
[67.0, 4.41, 5507.55],
[59.0, 2.89, 5223.69],
[56.0, 2.17, 155.42],
[45.0, 0.4, 796.3],
[36.0, 0.47, 775.52],
[29.0, 2.65, 7.11],
[21.0, 5.34, 0.98],
[19.0, 1.85, 5486.78],
[19.0, 4.97, 213.3],
[17.0, 2.99, 6275.96],
[16.0, 0.03, 2544.31],
[16.0, 1.43, 2146.17],
[15.0, 1.21, 10977.08],
[12.0, 2.83, 1748.02],
[12.0, 3.26, 5088.63],
[12.0, 5.27, 1194.45],
[12.0, 2.08, 4694.0],
[11.0, 0.77, 553.57],
[10.0, 1.3, 6286.6],
[10.0, 4.24, 1349.87],
[9.0, 2.7, 242.73],
[9.0, 5.64, 951.72],
[8.0, 5.3, 2352.87],
[6.0, 2.65, 9437.76],
[6.0, 4.67, 4690.48],
]
)
L2_TABLE = np.array(
[
[52919.0, 0.0, 0.0],
[8720.0, 1.0721, 6283.0758],
[309.0, 0.867, 12566.152],
[27.0, 0.05, 3.52],
[16.0, 5.19, 26.3],
[16.0, 3.68, 155.42],
[10.0, 0.76, 18849.23],
[9.0, 2.06, 77713.77],
[7.0, 0.83, 775.52],
[5.0, 4.66, 1577.34],
[4.0, 1.03, 7.11],
[4.0, 3.44, 5573.14],
[3.0, 5.14, 796.3],
[3.0, 6.05, 5507.55],
[3.0, 1.19, 242.73],
[3.0, 6.12, 529.69],
[3.0, 0.31, 398.15],
[3.0, 2.28, 553.57],
[2.0, 4.38, 5223.69],
[2.0, 3.75, 0.98],
]
)
L3_TABLE = np.array(
[
[289.0, 5.844, 6283.076],
[35.0, 0.0, 0.0],
[17.0, 5.49, 12566.15],
[3.0, 5.2, 155.42],
[1.0, 4.72, 3.52],
[1.0, 5.3, 18849.23],
[1.0, 5.97, 242.73],
]
)
L4_TABLE = np.array([[114.0, 3.142, 0.0], [8.0, 4.13, 6283.08], [1.0, 3.84, 12566.15]])
L5_TABLE = np.array([[1.0, 3.14, 0.0]])
HELIO_L = [L0_TABLE, L1_TABLE, L2_TABLE, L3_TABLE, L4_TABLE, L5_TABLE]
# heliocentric latitude coefficients
B0_TABLE = np.array(
[
[280.0, 3.199, 84334.662],
[102.0, 5.422, 5507.553],
[80.0, 3.88, 5223.69],
[44.0, 3.7, 2352.87],
[32.0, 4.0, 1577.34],
]
)
B1_TABLE = np.array([[9.0, 3.9, 5507.55], [6.0, 1.73, 5223.69]])
HELIO_B = [B0_TABLE, B1_TABLE]
# heliocentric radius coefficients
R0_TABLE = np.array(
[
[100013989.0, 0.0, 0.0],
[1670700.0, 3.0984635, 6283.07585],
[13956.0, 3.05525, 12566.1517],
[3084.0, 5.1985, 77713.7715],
[1628.0, 1.1739, 5753.3849],
[1576.0, 2.8469, 7860.4194],
[925.0, 5.453, 11506.77],
[542.0, 4.564, 3930.21],
[472.0, 3.661, 5884.927],
[346.0, 0.964, 5507.553],
[329.0, 5.9, 5223.694],
[307.0, 0.299, 5573.143],
[243.0, 4.273, 11790.629],
[212.0, 5.847, 1577.344],
[186.0, 5.022, 10977.079],
[175.0, 3.012, 18849.228],
[110.0, 5.055, 5486.778],
[98.0, 0.89, 6069.78],
[86.0, 5.69, 15720.84],
[86.0, 1.27, 161000.69],
[65.0, 0.27, 17260.15],
[63.0, 0.92, 529.69],
[57.0, 2.01, 83996.85],
[56.0, 5.24, 71430.7],
[49.0, 3.25, 2544.31],
[47.0, 2.58, 775.52],
[45.0, 5.54, 9437.76],
[43.0, 6.01, 6275.96],
[39.0, 5.36, 4694.0],
[38.0, 2.39, 8827.39],
[37.0, 0.83, 19651.05],
[37.0, 4.9, 12139.55],
[36.0, 1.67, 12036.46],
[35.0, 1.84, 2942.46],
[33.0, 0.24, 7084.9],
[32.0, 0.18, 5088.63],
[32.0, 1.78, 398.15],
[28.0, 1.21, 6286.6],
[28.0, 1.9, 6279.55],
[26.0, 4.59, 10447.39],
]
)
R1_TABLE = np.array(
[
[103019.0, 1.10749, 6283.07585],
[1721.0, 1.0644, 12566.1517],
[702.0, 3.142, 0.0],
[32.0, 1.02, 18849.23],
[31.0, 2.84, 5507.55],
[25.0, 1.32, 5223.69],
[18.0, 1.42, 1577.34],
[10.0, 5.91, 10977.08],
[9.0, 1.42, 6275.96],
[9.0, 0.27, 5486.78],
]
)
R2_TABLE = np.array(
[
[4359.0, 5.7846, 6283.0758],
[124.0, 5.579, 12566.152],
[12.0, 3.14, 0.0],
[9.0, 3.63, 77713.77],
[6.0, 1.87, 5573.14],
[3.0, 5.47, 18849.23],
]
)
R3_TABLE = np.array([[145.0, 4.273, 6283.076], [7.0, 3.92, 12566.15]])
R4_TABLE = np.array([[4.0, 2.56, 6283.08]])
HELIO_R = [R0_TABLE, R1_TABLE, R2_TABLE, R3_TABLE, R4_TABLE]
# longitude and obliquity nutation coefficients
NUTATION_ABCD_ARRAY = np.array(
[
[-171996, -174.2, 92025, 8.9],
[-13187, -1.6, 5736, -3.1],
[-2274, -0.2, 977, -0.5],
[2062, 0.2, -895, 0.5],
[1426, -3.4, 54, -0.1],
[712, 0.1, -7, 0],
[-517, 1.2, 224, -0.6],
[-386, -0.4, 200, 0],
[-301, 0, 129, -0.1],
[217, -0.5, -95, 0.3],
[-158, 0, 0, 0],
[129, 0.1, -70, 0],
[123, 0, -53, 0],
[63, 0, 0, 0],
[63, 0.1, -33, 0],
[-59, 0, 26, 0],
[-58, -0.1, 32, 0],
[-51, 0, 27, 0],
[48, 0, 0, 0],
[46, 0, -24, 0],
[-38, 0, 16, 0],
[-31, 0, 13, 0],
[29, 0, 0, 0],
[29, 0, -12, 0],
[26, 0, 0, 0],
[-22, 0, 0, 0],
[21, 0, -10, 0],
[17, -0.1, 0, 0],
[16, 0, -8, 0],
[-16, 0.1, 7, 0],
[-15, 0, 9, 0],
[-13, 0, 7, 0],
[-12, 0, 6, 0],
[11, 0, 0, 0],
[-10, 0, 5, 0],
[-8, 0, 3, 0],
[7, 0, -3, 0],
[-7, 0, 0, 0],
[-7, 0, 3, 0],
[-7, 0, 3, 0],
[6, 0, 0, 0],
[6, 0, -3, 0],
[6, 0, -3, 0],
[-6, 0, 3, 0],
[-6, 0, 3, 0],
[5, 0, 0, 0],
[-5, 0, 3, 0],
[-5, 0, 3, 0],
[-5, 0, 3, 0],
[4, 0, 0, 0],
[4, 0, 0, 0],
[4, 0, 0, 0],
[-4, 0, 0, 0],
[-4, 0, 0, 0],
[-4, 0, 0, 0],
[3, 0, 0, 0],
[-3, 0, 0, 0],
[-3, 0, 0, 0],
[-3, 0, 0, 0],
[-3, 0, 0, 0],
[-3, 0, 0, 0],
[-3, 0, 0, 0],
[-3, 0, 0, 0],
]
)
NUTATION_YTERM_ARRAY = np.array(
[
[0, 0, 0, 0, 1],
[-2, 0, 0, 2, 2],
[0, 0, 0, 2, 2],
[0, 0, 0, 0, 2],
[0, 1, 0, 0, 0],
[0, 0, 1, 0, 0],
[-2, 1, 0, 2, 2],
[0, 0, 0, 2, 1],
[0, 0, 1, 2, 2],
[-2, -1, 0, 2, 2],
[-2, 0, 1, 0, 0],
[-2, 0, 0, 2, 1],
[0, 0, -1, 2, 2],
[2, 0, 0, 0, 0],
[0, 0, 1, 0, 1],
[2, 0, -1, 2, 2],
[0, 0, -1, 0, 1],
[0, 0, 1, 2, 1],
[-2, 0, 2, 0, 0],
[0, 0, -2, 2, 1],
[2, 0, 0, 2, 2],
[0, 0, 2, 2, 2],
[0, 0, 2, 0, 0],
[-2, 0, 1, 2, 2],
[0, 0, 0, 2, 0],
[-2, 0, 0, 2, 0],
[0, 0, -1, 2, 1],
[0, 2, 0, 0, 0],
[2, 0, -1, 0, 1],
[-2, 2, 0, 2, 2],
[0, 1, 0, 0, 1],
[-2, 0, 1, 0, 1],
[0, -1, 0, 0, 1],
[0, 0, 2, -2, 0],
[2, 0, -1, 2, 1],
[2, 0, 1, 2, 2],
[0, 1, 0, 2, 2],
[-2, 1, 1, 0, 0],
[0, -1, 0, 2, 2],
[2, 0, 0, 2, 1],
[2, 0, 1, 0, 0],
[-2, 0, 2, 2, 2],
[-2, 0, 1, 2, 1],
[2, 0, -2, 0, 1],
[2, 0, 0, 0, 1],
[0, -1, 1, 0, 0],
[-2, -1, 0, 2, 1],
[-2, 0, 0, 0, 1],
[0, 0, 2, 2, 1],
[-2, 0, 2, 0, 1],
[-2, 1, 0, 2, 1],
[0, 0, 1, -2, 0],
[-1, 0, 1, 0, 0],
[-2, 1, 0, 0, 0],
[1, 0, 0, 0, 0],
[0, 0, 1, 2, 0],
[0, 0, -2, 2, 2],
[-1, -1, 1, 0, 0],
[0, 1, 1, 0, 0],
[0, -1, 1, 2, 2],
[2, -1, -1, 2, 2],
[0, 0, 3, 2, 2],
[2, -1, 0, 2, 2],
]
)
# END COEFF
# now, we write to the actual code
DELTA_T = 67
@jit
def helio_vector(vec, jme):
"""This function calculates equation 9 across the vector"""
return np.sum(
vec[:, 0] * np.cos(vec[:, 1] + vec[:, 2] * jme[..., np.newaxis]), axis=-1
)
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def solar_position(timestamp, latitude, longitude, elevation):
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"""Calculate the position of the sun at a given location and time.
Args:
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timestamp (array-like): The timestamp(s) at each point.
latitude (array-like): The latitude(s) of each point.
longitude (array-like): The longitude(s) of each point.
elevation (array-like): The elevation of each point.
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Returns:
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ndarray: An array containing the altitude and azimuth for each point.
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"""
jd = timestamp / 86400.0 + 2440587.5
jc = (jd - 2451545) / 36525
jde = jd + DELTA_T / 86400.0
jce = (jde - 2451545) / 36525
jm = jc / 10
jme = jce / 10
# todo: make more elegant?
# TODO: vectorize later? it's kinda complex
# heliocentric longitude
l_rad = np.zeros_like(timestamp)
for idx, vec in enumerate(HELIO_L):
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l_rad = l_rad + helio_vector(vec, jme) * jme ** idx
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l_rad = l_rad / 10e8
l_deg = np.rad2deg(l_rad) % 360
# heliocentric latitude
b_rad = np.zeros_like(timestamp)
for idx, vec in enumerate(HELIO_B):
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b_rad = b_rad + helio_vector(vec, jme) * jme ** idx
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b_rad = b_rad / 10e8
b_deg = np.rad2deg(b_rad) % 360
# heliocentric radius
r_rad = np.zeros_like(timestamp)
for idx, vec in enumerate(HELIO_R):
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r_rad = r_rad + helio_vector(vec, jme) * jme ** idx
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r_rad = r_rad / 10e8
r_deg = np.rad2deg(r_rad) % 360
theta = (l_deg + 180) % 360
beta = -1 * b_deg
def cubic_poly(a, b, c, d):
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return a + b * jce + c * jce ** 2 + (jce ** 3) / d
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X0 = cubic_poly(297.85036, 445267.111480, -0.0019142, 189474)
X1 = cubic_poly(357.52772, 35999.050340, -0.0001603, -300000)
X2 = cubic_poly(134.96298, 477198.867398, 0.0086972, 56250)
X3 = cubic_poly(93.27191, 483202.017538, -0.0036825, 327270)
X4 = cubic_poly(125.04452, 1934.136261, 0.0020708, 450000)
X = np.vstack([X0, X1, X2, X3, X4]).T
nut = NUTATION_ABCD_ARRAY
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# TODO: these are gross - use loops instead of broadcasting?
# FIXME: use guvectorize, treat jce as a scalar.
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d_psi = (nut[:, 0] + jce[..., np.newaxis] * nut[:, 1]) * np.sin(
np.sum(X[:, np.newaxis, :] * NUTATION_YTERM_ARRAY[np.newaxis, ...], axis=2)
)
d_psi = np.sum(d_psi, axis=-1) / 36, 000, 000
d_epsilon = (nut[:, 2] + jce[..., np.newaxis] * nut[:, 3]) * np.cos(
np.sum(X[:, np.newaxis, :] * NUTATION_YTERM_ARRAY[np.newaxis, ...], axis=2)
)
d_epsilon = np.sum(d_epsilon, axis=-1) / 36, 000, 000
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u = jme[:, np.newaxis] / 10 * np.arange(0, 10).reshape((1, -1))
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epsilon_0 = np.array(
[
84381.448,
-4680.93,
1.55,
1999.25,
-51.38,
-249.67,
-39.05,
7.12,
27.87,
5.79,
2.45,
]
)
epsilon = np.sum(u * epsilon_0, axis=-1) / 3600 + d_epsilon
d_tau = -20.4898 / (3600 * r_deg)
sun_longitude = theta + d_psi + d_tau
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v_0 = 280.46061837 + 360.98564736629 * (jd - 2451545) + 0.000387933 * jc ** 2 - jc ** 3 / 38710000
v_0 = v_0 % 360
v = v_0 + d_psi * np.cos(np.deg2rad(epsilon))
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alpha = np.arctan2(np.sin(np.deg2rad(sun_longitude)) *
np.cos(np.deg2rad(epsilon)) -
np.tan(np.deg2rad(beta)) *
np.sin(np.deg2rad(epsilon)),
np.cos(np.deg2rad(sun_longitude)))
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alpha_deg = np.rad2deg(alpha) % 360
delta = np.arcsin(
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np.sin(np.deg2rad(beta)) *
np.cos(np.deg2rad(epsilon)) +
np.cos(np.deg2rad(beta)) *
np.sin(np.deg2rad(epsilon)) *
np.cos(np.deg2rad(sun_longitude))
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)
delta_deg = np.rad2deg(delta) % 360
h = v + latitude - alpha_deg
xi_deg = 8.794 / (3600 * r_deg)
u = np.arctan(0.99664719 * np.tan(latitude))
x = np.cos(u) + elevation / 6378140 * np.cos(latitude)
y = 0.99664719 * np.sin(u) + elevation / 6378140 * np.sin(latitude)
d_alpha = np.arctan2(-1 * x * np.sin(np.radians(xi_deg)) * np.sin(np.radians(h)), np.cos(delta))
d_alpha = np.rad2deg(d_alpha)
alpha_prime = alpha_deg + d_alpha
delta_prime = np.arctan2((np.sin(delta) - y * np.sin(np.radians(xi_deg))) * np.cos(np.radians(d_alpha)),
np.cos(delta) - x * np.sin(np.radians(xi_deg)) * np.cos(np.radians(h)))
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h_prime = h - d_alpha
e_0 = np.arcsin(np.sin(latitude) * np.sin(delta) + np.cos(latitude) * np.cos(delta_prime))
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