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transform_functions.py
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transform_functions.py
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# tranform_functions
# Jeffrey Mays
import numpy as np
import math as ma
RTOD = 180.0 / np.pi
DTOR = np.pi / 180.0
def skew(a):
# Skew semetric matrix
#
# INPUTS:
# a 3 by 1 vector of numbers
#
# OUTPUTS
# A 3 by 3 skew matrix
A = np.array([[0.0, -a[2], a[1]],
[a[2], 0.0, -a[0]],
[-a[1], a[0], 0.0]])
return A
def RPY(eul):
# Euler angle Diff EQ
#
# INPUTS:
# eul 3 by 1 set of Euler angles (phi, theta, psi) (rad)
#
# OUTPUTS
# A 3 by 3 matrix for Euler angle integration
sin_phi = np.sin(eul[0])
cos_phi = np.cos(eul[0])
sin_theta = np.sin(eul[1])
cos_theta = np.cos(eul[1])
A = np.array([[1.0, sin_phi*sin_theta, cos_phi*sin_theta],
[0.0, cos_phi*cos_theta, -sin_phi*cos_theta],
[0.0, sin_phi, cos_phi]])
A = 1.0/cos_theta*A
return A
def CTM_to_Euler(C):
# Compute Euler angles from transform matrix
#
# INPUTS:
# C 3 by 3 matrix C_b_n
#
# OUTPUTS
# Euler Euler angle vector (phi, theta, psi) (rad)
C = C.T # change input so these are correct
phi = float(ma.atan2(C[1, 2], C[2, 2]))
theta = float(-ma.asin(C[0, 2]))
psi = float(ma.atan2(C[0, 1], C[0, 0]))
Euler = np.zeros((1, 3))
Euler = np.array([phi, theta, psi])
return Euler
def Euler_to_CTM(eul):
# Create Coordinate transform matrix
#
# INPUTS:
# eul Euler angle vector (phi, theta, psi) (rad)
#
# OUTPUTS
# C.T Body to Nav Rotation Matrix (no transpose is opposite)
sin_phi = float(np.sin(eul[0]))
cos_phi = float(np.cos(eul[0]))
sin_theta = float(np.sin(eul[1]))
cos_theta = float(np.cos(eul[1]))
sin_psi = float(np.sin(eul[2]))
cos_psi = float(np.cos(eul[2]))
C = np.zeros((3, 3))
C[0, 0] = cos_theta * cos_psi
C[0, 1] = cos_theta * sin_psi
C[0, 2] = -sin_theta
C[1, 0] = -cos_phi * sin_psi + sin_phi * sin_theta * cos_psi
C[1, 1] = cos_phi * cos_psi + sin_phi * sin_theta * sin_psi
C[1, 2] = sin_phi * cos_theta
C[2, 0] = sin_phi * sin_psi + cos_phi * sin_theta * cos_psi
C[2, 1] = -sin_phi * cos_psi + cos_phi * sin_theta * sin_psi
C[2, 2] = cos_phi * cos_theta
return C.T # body to NED
def LLA_to_NED(lat, long, alt, x_ECEF, x_ref):
# LLA to NED using reference
#
# INPUTS:
# lat Lattitude (rad)
# long Longitude (rad)
# alt altitude (m)
# x_ECEF Position of interest in ECEF frame (m)
# x_ref Reference ECEF position for NED (m)
#
# OUTPUTS
# x_NED Position in NED frame based on x_ref
R_NED_to_ECEF = np.array([[-np.sin(lat)*np.cos(long),
-np.sin(long),
-np.cos(lat)*np.cos(long)],
[-np.sin(lat)*np.sin(long),
np.cos(long),
-np.cos(lat)*np.sin(long)],
[np.cos(lat),
0.0,
-np.sin(lat)]])
residual = np.array([[float(x_ECEF[0])-float(x_ref[0])],
[float(x_ECEF[1])-float(x_ref[1])],
[float(x_ECEF[2])-float(x_ref[2])]])
x_NED = np.matmul(R_NED_to_ECEF.T, residual)
return(x_NED)
def ECEF_to_LLA(r_eb_e):
# ECEF to LLA frame
#
# INPUTS:
# r_eb_e ECEF frame coordinate (m)
#
# OUTPUTS
# lat Lattitude (rad)
# long Longitude (rad)
# alt altitude (m)
f = 1.0/298.257223563
R0 = 6378137.0
e = np.sqrt(f*(2.0-f))
R_P = R0*(1.0-f)
long = ma.atan2(r_eb_e[1], r_eb_e[0])
p = np.sqrt(r_eb_e[0]**2+r_eb_e[1]**2)
E = np.sqrt(R0**2-R_P**2)
F = 54.0*(R_P*r_eb_e[2])**2
G = p**2+(1.0-e**2)*r_eb_e[2]**2-(e*E)**2
c = e**4*F*p**2/(G**3)
s = (1.0+c+np.sqrt(c**2+2*c))**(1.0/3.0)
P = (F/(3*G**2))/((s+1.0/s+1.0)**2)
Q = np.sqrt(1.0+2.0*e**4*P)
k1 = (-P*e**2*p)/(1.0+Q)
k2 = 0.5*R0**2*(1.0+1.0/Q)
k3 = -P*(1.0-e**2)*(r_eb_e[2]**2)/(Q*(1.0+Q))
k4 = -0.5*P*p**2
k5 = p-e**2*(k1+np.sqrt(k2+k3+k4))
U = np.sqrt(k5**2+r_eb_e[2]**2)
V = np.sqrt(k5**2+(1.0-e**2)*r_eb_e[2]**2)
alt = U*(1.0-R_P**2/(R0*V))
z0 = (R_P**2*r_eb_e[2])/(R0*V)
ep = R0*e/R_P
lat = ma.atan((r_eb_e[2]+z0*ep**2)/p)
return lat, long, alt
def LLA_to_ECEF(lat, long, alt):
# LLA to ECEF frame
#
# INPUTS:
# lat Lattitude (rad)
# long Longitude (rad)
# alt altitude (m)
#
# OUTPUTS
# x ECEF x position (m)
# y ECEF y position (m)
# z ECEF z position (m)
R0 = 6378137.0 # WGS84 Equatorial radius in meters
f = 1/298.257223563
e = np.sqrt(f*(2.0 - f)) # WGS84 eccentricity
# calculate transverse radius of curvature (2.105)
R_E = R0 / np.sqrt(1.0-(e*np.sin(lat))**2)
# Convert position using (2.112)
cos_lat = np.cos(lat)
sin_lat = np.sin(lat)
cos_long = np.cos(long)
sin_long = np.sin(long)
r_eb_e = np.array([[(R_E + alt) * cos_lat * cos_long],
[(R_E + alt) * cos_lat * sin_long],
[((1.0 - e**2) * R_E + alt) * sin_lat]])
x = r_eb_e[0]
y = r_eb_e[1]
z = r_eb_e[2]
return x, y, z
def ECEF_to_NED(x_ecef, latR, longR, alt):
# ECEF to NED frame (Using differnt method)
#
# INPUTS:
# x_ecef ECEF position of interest (m)
# lat Lattitude (rad)
# long Longitude (rad)
# alt altitude (m)
#
# OUTPUTS
# NED NED position based on LLA coordinates (m)
x, y, z = LLA_to_ECEF(latR, longR, alt)
ref_e = np.array([x, y, z])
delta_E = np.zeros((3, 1))
delta_E[0] = x_ecef[0] - ref_e[0]
delta_E[1] = x_ecef[1] - ref_e[1]
delta_E[2] = x_ecef[2] - ref_e[2]
ENU = np.zeros((3, 1))
ENU[0] = -np.sin(longR)*delta_E[0] + np.cos(longR)*delta_E[1]
ENU[1] = -np.sin(latR)*np.cos(longR)*delta_E[0] - np.sin(latR)*np.sin(longR)*delta_E[1] + np.cos(latR)*delta_E[2]
ENU[2] = -np.cos(latR)*np.cos(longR)*delta_E[0] + np.cos(latR)*np.sin(longR)*delta_E[1] + np.sin(latR)*delta_E[2]
C = np.zeros((3, 3))
C[0, 1] = 1.0 # ENU to NED Rotation Matrix
C[1, 0] = 1.0
C[2, 2] = -1.0
NED = np.zeros((3, 1))
NED = C@ENU
return NED
def ECI_to_ECEF(xp, yp, Omega, i):
# ECI to ECEF frame
#
# INPUTS:
# xp
# yp
# Omega
# i
#
# OUTPUTS
# r_eb_e ECEF frame coordinate (m)
x = xp*ma.cos(Omega) - yp*ma.cos(i)*ma.sin(Omega)
y = xp*ma.sin(Omega) + yp*ma.cos(i)*ma.cos(Omega)
z = yp*ma.sin(i)
r_eb_e = np.array([x, y, z])
return r_eb_e