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OpenVerne.py
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OpenVerne.py
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# -*- coding: utf-8 -*-
# Copyright (c) 2017 Interstellar Technologies Inc. All Rights Reserved.
# Authors : Takahiro Inagawa
#
# Lisence : MIT Lisence
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
#
# The above copyright notice and this permission notice shall be included in all
# copies or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
# SOFTWARE.
# ==============================================================================
"""
OpenVerne.IIP - rocket or missile Instantaneous Impact Point(IIP) calculation
The instantaneous impact point (IIP) of a rocket, given its position and velocity,
is defined as its touchdown point(altitude=0[m]) assuming a free-fall flight (without propulsion).
The IIP is considered as a very important information for safe launch operation
of a rocket.
cf.
Jaemyung Ahn and Woong-Rae Roh. "Noniterative Instantaneous Impact Point Prediction Algorithm for Launch Operations",
Journal of Guidance, Control, and Dynamics, Vol. 35, No. 2 (2012), pp. 645-648.
https://doi.org/10.2514/1.56395
"""
import numpy as np
from numpy import cos, sin, tan, arcsin, arctan2, arccos
from numpy import sqrt, deg2rad, rad2deg, pi
class WGS84():
def __init__(self):
self.re_a = 6378137.0 # [m] WGS84の長軸
self.eccen1 = 8.1819190842622e-2 # First Eccentricity
self.eccen1sqr = 6.69437999014e-3 # First Eccentricity squared
self.one_f = 298.257223563 # 扁平率fの1/f（平滑度）
self.re_b = 6356752.314245 # [m] WGS84の短軸
self.e2 = 6.6943799901414e-3 # 第一離心率eの2乗
self.ed2 = 6.739496742276486e-3 # 第二離心率e'の2乗
wgs84 = WGS84()
omega_earth = 7.2921159e-5; # 地球の自転角速度[rad/s]
def posECEF_from_LLH(posLLH_):
"""
Args:
posLLH_ (np.array 3x1) : position in LLH coordinate [deg, deg, m]
Return:
(np.array 3x1) : position in ECEF coordinate [m, m, m]
"""
lat = deg2rad(posLLH_[0])
lon = deg2rad(posLLH_[1])
alt = posLLH_[2]
W = sqrt(1.0 - wgs84.e2 * sin(lat) * sin(lat))
N = wgs84.re_a / W
pos0 = (N + alt) * cos(lat) * cos(lon)
pos1 = (N + alt) * cos(lat) * sin(lon)
pos2 = (N * (1 - wgs84.e2) + alt) * sin(lat)
return np.array([pos0, pos1, pos2])
def dcmECI2ECEF(second):
"""
Args:
second (double) : time from refarence time[s]
Return:
dcm (np.array 3x3) : DCM from ECI to ECEF
"""
theta = omega_earth * second
dcm = np.array([[cos(theta), sin(theta), 0.0],
[-sin(theta), cos(theta), 0.0],
[0.0, 0.0, 1.0]])
return dcm
def posECI(posECEF_, second):
"""
Args:
posECEF_ (np.array 3x1) : position in ECEF coordinate [m, m, m]
second (double) : time from refarence time [s]
Return:
(np.array 3x1) : position in ECI coordinate [m, m, m]
"""
dcmECI2ECEF_ = dcmECI2ECEF(second)
dcmECEF2ECI_ = dcmECI2ECEF_.T
return dcmECEF2ECI_.dot(posECEF_)
def dcmECEF2NED(posLLH_):
"""
Args:
posLLH_ (np.array 3x1) : [deg, deg, m]
Return:
dcm (np.array 3x3) : DCM from ECEF to NED
"""
lat = deg2rad(posLLH_[0])
lon = deg2rad(posLLH_[1])
dcm = np.array([[-sin(lat)*cos(lon), -sin(lat)*sin(lon), cos(lat)],
[-sin(lon), cos(lon), 0],
[-cos(lat)*cos(lon), -cos(lat)*sin(lon), -sin(lat)]])
return dcm
def dcmECI2NED(dcmECEF2NED_, dcmECI2ECEF_):
return dcmECEF2NED_.dot(dcmECI2ECEF_)
def posLLH(posECEF_):
"""
Args:
posECEF_ (np.array 3x1) : [deg, deg, m]
Return:
(np.array 3x1) : position in LLH coordinate [deg, deg, m]
"""
def n_posECEF2LLH(phi_n_deg):
return wgs84.re_a / sqrt(1.0 - wgs84.e2 * sin(deg2rad(phi_n_deg)) * sin(deg2rad(phi_n_deg)))
p = sqrt(posECEF_[0] **2 + posECEF_[1] **2)
theta = arctan2(posECEF_[2] * wgs84.re_a, p * wgs84.re_b) # rad
lat = rad2deg(arctan2(posECEF_[2] + wgs84.ed2 * wgs84.re_b * pow(sin(theta), 3), p - wgs84.e2 * wgs84.re_a * pow(cos(theta),3)))
lon = rad2deg(arctan2(posECEF_[1], posECEF_[0]))
alt = p / cos(deg2rad(lat)) - n_posECEF2LLH(lat)
return np.array([lat, lon, alt])
def lat_from_radius(radius):
""" Return latitude[deg] from earth radius
Args:
radius (double) : earth radius[m]
Return:
(double) : latitude [deg]
"""
lat_rad = arcsin(sqrt(1/wgs84.e2 * (1 - (radius**2) / (wgs84.re_a**2))))
return rad2deg(lat_rad)
class IIP:
def __init__(self, posLLH_, velNED_):
""" calculate IIP from current position(LLH) & current velocity(NED)
Args:
posLLH_ (np.array 3x1) : position at current point(LLH) [deg, deg, m]
velNED_ (np.array 3x1) : velocity at current point(NED frame) [m/s, m/s, m/s]
Attributes:
posLLH_IIP_deg (np.array 2x1) : IIP position (LLH coordinate) [deg, deg]
posLLH_IIP_rad (np.array 2x1) : IIP position (LLH coordinate) [rad, rad]
distance_ECEF (double) : earth surface distance from current point to IIP [m]
tf (double) : time of flight from current point to IIP [s]
Usage:
> _IIP = IIP(posLLH_, velNED_)
> print(_IIP)
"""
earth_radius = wgs84.re_a # 地球半径 [m]
mu = 3.986004418 * 10**(14) # 地球重力定数 m3s-2
# 初期位置・速度のECI座標系への変換
self.posLLH_ = posLLH_
self.velNED_ = velNED_
posECI_init_ = posECI(posECEF_from_LLH(posLLH_), 0)
dcmECI2NED_ = dcmECI2NED(dcmECEF2NED(posLLH_), dcmECI2ECEF(0))
omegaECI2ECEF_ = np.array([[0.0, -omega_earth, 0.0],
[omega_earth, 0.0, 0.0],
[0.0, 0.0, 0.0]]) # 角速度テンソル
velECI_init_ = np.dot(dcmECI2NED_.transpose(), velNED_) + omegaECI2ECEF_.dot(posECI_init_)
# 計算に必要なr0, v0の絶対値と単位ベクトル、初期のγ:flight-path angleの計算
self.r0 = np.linalg.norm(posECI_init_)
self.v0 = np.linalg.norm(velECI_init_)
self.ir0 = posECI_init_ / np.linalg.norm(posECI_init_) # 位置の単位ベクトル
self.iv0 = velECI_init_ / np.linalg.norm(velECI_init_) # 速度の単位ベクトル
gamma0 = arcsin(np.dot(self.ir0, self.iv0)) # [rad]
self.gamma0 = gamma0 # gammaを外から見るためにインスタンス変数に
lam = self.v0**2 / (mu / self.r0) # lambda
self.lam = lam
def rp_calc(rp_temp):
""" 関数内関数 収束計算のためにrp入力して計算される緯度(lat)から計算されるrpの差分を出力
Args:
rp_temp (double) : その緯度での地球半径 [m]
Return:
(doubel) : 計算されるrpと入力rpの差分
"""
# phi:flight angle of a rocket の計算
c1 = - tan(gamma0)
c2 = 1 - 1/(lam * cos(gamma0)**2)
c3 = self.r0 / rp_temp - 1 / (lam * cos(gamma0)**2)
c12 = c1 ** 2
c22 = c2 ** 2
c32 = c3 ** 2
try:
phi = arcsin((c1*c3 + sqrt(c12*c32 - (c12+c22)*(c32-c22))) / (c12 + c22))
except RuntimeWarning:
phi = np.nan
# IIPの位置の単位ベクトルとそこから計算されるECI座標系でのIIP緯度経度 参考：eq.(13)~(15)
self.ip = cos(gamma0 + phi)/cos(gamma0) * self.ir0 + sin(phi) / cos(gamma0) * self.iv0 # IIP単位ベクトル(ECI)
self.ip = self.ip / np.linalg.norm(self.ip)
IIP_LLH_deg = posLLH(self.ip * rp_temp)
lat_ECI_IIP_rad = deg2rad(IIP_LLH_deg[0])
lon_ECI_IIP_rad = deg2rad(IIP_LLH_deg[1])
# print("phi = %3f [deg], lat IIP %.3f [deg]" % (rad2deg(phi),rad2deg(lat_ECI_IIP_rad)))
rp_new = wgs84.re_a * sqrt(1 - wgs84.e2 * sin(lat_ECI_IIP_rad)**2)
return rp_temp - rp_new
rp1 = wgs84.re_b # 収束計算のための二分法の下区間 地球短半径
rp2 = wgs84.re_a # 上区間 地球長半径
epsilon = 1e-3 # 収束計算の収束誤差
while True: # 二分法
rpM = (rp1 + rp2) / 2
hantei_rp1 = rp_calc(rp1) # 正か負かnan
hantei_rp2 = rp_calc(rp2) # 正か負かnan
hantei_rpM = rp_calc(rpM)
# print("rpM = %.1f, 1:%.5f, 2:%.5f, M:%.5f" % (rpM, hantei_rp1, hantei_rp2, hantei_rpM))
if(hantei_rpM < 0):
rp1 = rpM
else:
rp2 = rpM
if (abs(hantei_rpM) < epsilon):
break
# 収束したrpの値をrpとして設定
rp = rpM
# phi:flight angle of a rocket の計算
c1 = - tan(gamma0)
c2 = 1 - 1/(lam * cos(gamma0)**2)
c3 = self.r0 / rp - 1 / (lam * cos(gamma0)**2)
c12 = c1 ** 2
c22 = c2 ** 2
c32 = c3 ** 2
phi = arcsin((c1*c3 + sqrt(c12*c32 - (c12+c22)*(c32-c22))) / (c12 + c22))
self.phi = phi
# IIPの位置の単位ベクトルとそこから計算されるECI座標系でのIIP緯度経度 参考：eq.(13)~(15)
self.ip = cos(gamma0 + phi)/cos(gamma0) * self.ir0 + sin(phi) / cos(gamma0) * self.iv0 # IIP単位ベクトル(ECI)
self.ip = self.ip / np.linalg.norm(self.ip)
IIP_LLH_deg = posLLH(self.ip * rp)
lat_ECI_IIP_rad = deg2rad(IIP_LLH_deg[0])
lon_ECI_IIP_rad = deg2rad(IIP_LLH_deg[1])
posLLH_ECI_IIP_rad = np.array([lat_ECI_IIP_rad, lon_ECI_IIP_rad, 0])
posLLH_init_rad = np.zeros(3)
posLLH_init_rad[0] = deg2rad(posLLH_[0])
posLLH_init_rad[1] = deg2rad(posLLH_[1])
# 初期位置からIIPまでの地球表面距離
# cf. https://keisan.casio.jp/exec/system/1257670779
self.distance_ECI = earth_radius * arccos(sin(posLLH_init_rad[0])*sin(posLLH_ECI_IIP_rad[0]) + cos(posLLH_init_rad[0])*cos(posLLH_ECI_IIP_rad[0])*cos(posLLH_ECI_IIP_rad[1]-posLLH_init_rad[1]))
# 飛翔時間の計算 参考:eq.(19)
t1 = self.r0 / self.v0 / cos(gamma0)
t2 = tan(gamma0) * (1 - cos(phi)) + (1 - lam) * sin(phi)
t3 = (2 - lam) * ((1 - cos(phi)) / (lam * cos(gamma0)**2))
t4 = (2 - lam) * (cos(gamma0 + phi) / cos(gamma0))
t5 = 2 * cos(gamma0) / (lam * (2 / lam - 1)**1.5)
t6u = sqrt(2 / lam - 1)
t6l = cos(gamma0) * tan(pi/2 - phi/2) - sin(gamma0)
self.tf = t1 * ((t2 / (t3 + t4)) + t5 * arctan2(t6u, t6l))
# 飛翔時間より、地球自転を考慮し、落下位置の緯度経度算出 参考：eq(14),(15)
lat_ECEF_IIP_rad = lat_ECI_IIP_rad
lon_ECEF_IIP_rad = lon_ECI_IIP_rad - omega_earth * self.tf
self.posLLH_IIP_rad = np.array([lat_ECEF_IIP_rad, lon_ECEF_IIP_rad])
self.posLLH_IIP_deg = np.array([rad2deg(lat_ECEF_IIP_rad), rad2deg(lon_ECEF_IIP_rad)])
# 初期位置からIIPまでの地球表面距離
self.distance_ECEF = earth_radius * arccos(sin(posLLH_init_rad[0])*sin(self.posLLH_IIP_rad[0]) + cos(posLLH_init_rad[0])*cos(self.posLLH_IIP_rad[0])*cos(self.posLLH_IIP_rad[1]-posLLH_init_rad[1]))
def __repr__(self):
print("==== current point ====")
print("lat = %.6f [deg], lon = %.6f [deg]" % (self.posLLH_[0], self.posLLH_[1]))
print("altitude = %.1f [m]" %(self.posLLH_[2]))
print("velocity(NED) = %.1f [m/s], %.1f [m/s], %.1f [m/s]" % (self.velNED_[0], self.velNED_[1], self.velNED_[2]))
print("r0(ECI) = %.1f [m]" % (self.r0))
print("v0(ECI) = %.1f [m/s]" % (self.v0))
print("unit vector of r0 (ECI) = [%.6f, %.6f, %.6f]" % (self.ir0[0], self.ir0[1], self.ir0[2]))
print("unit vector of v0 (ECI) = [%.6f, %.6f, %.6f]" % (self.iv0[0], self.iv0[1], self.iv0[2]))
print("gamma0 = %.4f [deg]" % (rad2deg(self.gamma0)))
print("==== IIP (Instantaneous Impact Point) ====")
print("lat = %.6f [deg], lon = %.6f [deg]" % (self.posLLH_IIP_deg[0], self.posLLH_IIP_deg[1]))
print("distance of earth surface ECEF = %.1f [m]" % (self.distance_ECEF))
print("time of flight = %.2f [s]" % (self.tf))
print("distance of earth surface ECI = %.1f [m]" % (self.distance_ECI))
print("flight angle of a rocket = %.6f [deg]" % (rad2deg(self.phi)))
print("unit vector of IIP(ECI) = [%.6f, %.6f, %.6f]" % (self.ip[0], self.ip[1], self.ip[2]))
return ""
def disp(self):
""" Simple display of result """
print("==== current point ====")
print("lat = %.6f [deg], lon = %.6f [deg]" % (self.posLLH_[0], self.posLLH_[1]))
print("altitude = %.1f [m]" %(self.posLLH_[2]))
print("velocity(NED) = %.1f [m/s], %.1f [m/s], %.1f [m/s]" % (self.velNED_[0], self.velNED_[1], self.velNED_[2]))
print("==== IIP (Instantaneous Impact Point) ====")
print("lat = %.6f [deg], lon = %.6f [deg]" % (self.posLLH_IIP_deg[0], self.posLLH_IIP_deg[1]))
print("distance of earth surface = %.1f [m]" % (self.distance_ECEF))
print("time of flight = %.2f [s]" % (self.tf))
if __name__ == '__main__':
posLLH_ = np.array([40, 140, 100])
velNED_ = np.array([10, 0, 0])
_IIP = IIP(posLLH_, velNED_)
print(_IIP)