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AOAwithTDOA.py
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AOAwithTDOA.py
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"""
Pypredict
Orbit prediction software. Displays the satellites' position and
orbital parameters in real time. Simulates satellite localization
and deployment.
Copyright (C) 2018-2021, Matías Vidal Valladares, matvidal.
Authors: Matías Vidal Valladares <[email protected]>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
This python class contains methods to estimate the position of an
unknown source, such as a femto-satellite, given two known
positions, such as two CubeSats. We use the TDOA and AOA of a
simulated signal sent by the unknown source. This class replicates
the figures of the research paper called A simple and accurate
TDOA-AOA localization method using two stations. After this, we
study the performance of this method in space, where the base
stations are CubeSats which are not fixed in one position and their
location is affected by the accuracy of their GNSS devices. We also
simulate the deployment of a femto-satellite from one of these
CubeSats and evaluate the location estimation for different
directions of deployment, affected by the accuracy of the attitude
and control systems, and different deployment points in the
mother-CubeSat's orbit. All the research done in this class is for
a research paper submitted at Remote Sensing with the name:
A femto-satellite localization method based on TDOA and AOA using
two CubeSats.
"""
from numpy import arange, arctan2, arccos, array, cos, linalg, matrix, mean, pi, random, savetxt, sin, std, sqrt, zeros, transpose
from pypredict.dpl import Dpl
from pypredict.sat import Sat
from matplotlib.pyplot import axis, plot, setp, show, subplots, subplots_adjust, tight_layout, xticks
from datetime import datetime, timedelta
from pkg_resources import resource_filename
import csv
class Locate(object):
def __init__(self):
print("Initializing AOA with TDOA")
def __call__(self):
return self
def get_distance(self, u, s):
"""
Calculates the Euclidean distance between the vectors u and s.
Parameters
----------
u : numpy.matrix
Vector of the unknown source position (femto-satellite) in
meters.
s : numpy.matrix
Vector of known position (CubeSat) in meters.
Returns
-------
d
Distance between position vectors u and s.
"""
d = sqrt((u[0,0] - s[0,0])**2 + (u[1,0] - s[1,0])**2 + (u[2,0] - s[2,0])**2)
return d
def get_real_vector(self, u, s1, s2):
"""
Returns the TDOA and the two angles of arrival without
measurement noise, given the position vectors u, s1 and s2.
Parameters
----------
u : numpy.matrix
Vector of the unknown source position (femto-satellite) in
meters.
s1 : numpy.matrix
Vector of known position 1 (CubeSat 1) in meters.
s2 : numpy.matrix
Vector of known position 2 (CubeSat 2) in meters.
Returns
-------
d21
Difference between d2 (distance CubeSat2-Femto-satellite)
and d1 (distance Cubesat1-Femto-satellite) in meters.
theta1
Azimuth of the femto-satellite seen by known position 1
(CubeSat 1) in degrees.
phi1
Elevation of the femto-satellite seen by known position 1
(CubeSat 1) in degrees.
theta2
Azimuth of the femto-satellite seen by known position 2
(CubeSat 2) in degrees.
phi2
Elevation of the femto-satellite seen by known position 2
(CubeSat 2) in degrees.
"""
d1 = linalg.norm(u - s1)
d2 = linalg.norm(u - s2)
d21 = d2 - d1
theta1 = arctan2(u[1,0] - s1[1,0], u[0,0] - s1[0,0])
phi1 = arctan2(u[2,0] - s1[2,0], sqrt((u[0,0] - s1[0,0])**2 + (u[1,0] - s1[1,0])**2))
theta2 = arctan2(u[1,0] - s2[1,0], u[0,0] - s2[0,0])
phi2 = arctan2(u[2,0] - s2[2,0], sqrt((u[0,0] - s2[0,0])**2 + (u[1,0] - s2[1,0])**2))
return d21, theta1, phi1, theta2, phi2
def get_b(self, theta, phi):
"""
Returns the unit vector of the location of the femto-satellite,
at the angles theta and phi with respect to a given CubeSat.
Parameters
----------
theta : numpy.float64
Azimuth of the femto-satellite seen by a given known
position (CubeSat).
phi : numpy.float64
Elevation of the femto-satellite seen by a given known
position (CubeSat).
Returns
-------
b
Unit vector of the unknown source (femto-satellite)
position.
"""
cos_phi = cos(phi)
return matrix([[cos_phi*cos(theta)], [cos_phi*sin(theta)], [sin(phi)]])
def get_Gm(self, theta, phi):
"""
Parameters
----------
theta : numpy.float64
Azimuth of the femto-satellite seen by a given known
position (CubeSat).
phi : numpy.float64
Elevation of the femto-satellite seen by a given known
position (CubeSat).
Returns
-------
Gm
Matrix used to calculate G.
"""
cos_theta = cos(theta)
sin_phi = sin(phi)
sin_theta = sin(theta)
Gm = matrix([[sin_theta, sin_phi*cos_theta],
[-cos_theta, sin_phi*sin_theta],
[0, -cos(phi)]])
return Gm
def get_G(self, b1, b2, G1, G2):
"""
b1 : numpy.matrix
Unit vector of the unknown source (femto-satellite)
position with respect to position s1 (CubeSat 2).
b2 : numpy.matrix
Unit vector of the unknown source (femto-satellite)
position with respect to position s2 (CubeSat 2).
G1 : numpy.matrix
Matrix used to calculate G or h.
G2 : numpy.matrix
Matrix used to calculate G or h.
Returns
-------
G
Matrix that depends on the angle measurements.
"""
two_b2_minus_b1 = 2*(b2 - b1)
G = matrix([[two_b2_minus_b1[0,0], G1[0,0], G1[0,1], G2[0,0], G2[0,1]],
[two_b2_minus_b1[1,0], G1[1,0], G1[1,1], G2[1,0], G2[1,1]],
[two_b2_minus_b1[2,0], G1[2,0], G1[2,1], G2[2,0], G2[2,1]]])
return G
def get_h(self, d21, s1, s2, b1, b2, G1, G2):
"""
Parameters
----------
d21 : numpy.float64
Difference between d2 (distance CubeSat2-Femto-satellite)
and d1 (distance Cubesat1-Femto-satellite) in meters.
s1 : numpy.matrix
Vector of known position 1 (CubeSat 1) in meters.
s2 : numpy.matrix
Vector of known position 2 (CubeSat 2) in meters.
b1 : numpy.matrix
Unit vector of the unknown source (femto-satellite)
position with respect to position s1 (CubeSat 2).
b2 : numpy.matrix
Unit vector of the unknown source (femto-satellite)
position with respect to position s2 (CubeSat 2).
G1 : numpy.matrix
Matrix used to calculate G or h.
G2 : numpy.matrix
Matrix used to calculate G or h.
"""
s1TG1 = s1.transpose()*G1
s2TG2 = s2.transpose()*G2
h = matrix([[((b2 - b1).transpose()*(s1 + s2 - d21*b1))[0,0]],
[s1TG1[0,0]], [s1TG1[0,1]], [s2TG2[0,0]], [s2TG2[0,1]]])
return h
def get_Lm(self, theta, phi):
"""
Parameters
----------
theta : numpy.float64
Azimuth of the femto-satellite seen by a given known
position (CubeSat).
phi : numpy.float64
Elevation of the femto-satellite seen by a given known
position (CubeSat).
Returns
-------
Lm
Matrix used to calculate the matrix T.
"""
cos_phi = cos(phi)
cos_theta = cos(theta)
sin_phi = sin(phi)
sin_theta = sin(theta)
Lm = matrix([[-cos_phi*sin_theta, -sin_phi*cos_theta],
[cos_phi*cos_theta, -sin_phi*sin_theta],
[0, cos_phi]])
return Lm
def get_Tm(self, d, phi):
"""
Parameters
----------
d : numpy.float64
Distance between the unknown source (femto-satellite)
and one of the known positions (CubeSats).
phi : numpy.float64
Elevation of the femto-satellite seen by a given known
position (CubeSat).
Returns
-------
Tm
Matrix used to calculate the matrix T.
"""
Tm = -d*matrix([[cos(phi), 0],
[0, 1]])
return Tm
def get_T(self, d1, d2, b1, b2, L1, L2, T1, T2):
"""
Parameters
----------
d1 : numpy.float64
Distance between the unknown source (femto-satellite)
and the known position 1 (CubeSat 1).
d2 : numpy.float64
Distance between the unknown source (femto-satellite)
and the known position 2 (CubeSat 2).
b1 : numpy.matrix
Unit vector of the unknown source (femto-satellite)
position with respect to position s1 (CubeSat 2).
b2 : numpy.matrix
Unit vector of the unknown source (femto-satellite)
position with respect to position s2 (CubeSat 2).
L1 : numpy.matrix
Matrix in terms of the azimuth and elevation from the
known position 1 (CubeSat 1) used to calculate T.
L2 : numpy.matrix
Matrix in terms of the azimuth and elevation from the
known position 2 (CubeSat 2) used to calculate T.
L1 : numpy.matrix
Matrix in terms of the distance and elevation from
the known position 1 (CubeSat 1) used to calculate T.
L2 : numpy.matrix
Matrix in terms of the distance and elevation from
the known position 2 (CubeSat 2) used to calculate T.
Returns
-------
T
Matrix used to calculate the covariance matrix W.
"""
r1b2TL1 = d1*b2.transpose()*L1
r2b1TL2 = d2*b1.transpose()*L2
b2_m_b1_t = ((b2 - b1).transpose()*b1)[0,0]
T = matrix([[-b2_m_b1_t, r1b2TL1[0,0], r1b2TL1[0,1], -r2b1TL2[0,0], -r2b1TL2[0,1]],
[0.0, T1[0,0], T1[0,1], 0.0, 0.0],
[0.0, T1[1,0], T1[1,1], 0.0, 0.0],
[0.0, 0.0, 0.0, T2[0,0], T2[0,1]],
[0.0, 0.0, 0.0, T2[1,0], T2[1,1]]])
return T
def get_Q(self, std_RD, std_AOA):
"""
Generates the measurement noise covariance matrix.
Parameters
----------
std_RD : float
Standard deviation of the range difference in meters.
std_AOA : float
Standard deviation of the AOA in radians.
Returns
-------
Q
Measurement noise covariance matrix.
"""
Q = matrix([[std_RD**2, 0.0, 0.0, 0.0, 0.0],
[0.0, std_AOA**2, 0.0, 0.0, 0.0],
[0.0, 0.0, std_AOA**2, 0.0, 0.0],
[0.0, 0.0, 0.0, std_AOA**2, 0.0],
[0.0, 0.0, 0.0, 0.0, std_AOA**2]])
return Q
def get_W(self, Q, T):
"""
Returns the covariance matrix.
Parameters
----------
Q : numpy.matrix
Measurement noise covariance matrix.
T : numpy.matrix
Matrix used to calculate the covariance matrix W.
If this matrix is invertible, the method gives a unique
solution.
Returns
-------
W
The covariance matrix.
"""
W = T*Q*T.transpose()
return W
def get_error_vector(self, std_RD, std_AOA, L):
"""
Generates the measurement error for the TDOA and AOA.
Parameters
----------
std_RD : float
Standard deviation of the range difference in meters.
std_AOA : float
Standard deviation of the AOA in radians.
L : int
Number of ensemble runs.
Returns
-------
e
Error vector of the TDOA in range and two AOA pairs.
"""
e = random.randn(5, L)
e[0,:] = std_RD*(e[0,:] - mean(e[0,:]))
e[1,:] = std_AOA*(e[1,:] - mean(e[1,:]))
e[2,:] = std_AOA*(e[2,:] - mean(e[2,:]))
e[3,:] = std_AOA*(e[3,:] - mean(e[3,:]))
e[4,:] = std_AOA*(e[4,:] - mean(e[4,:]))
return e
def get_MSE(self, u, s1, s2, k, Q):
"""
Returns the mean-square error.
Parameters
----------
u : numpy.matrix
Vector of the unknown source position (femto-satellite) in
meters.
s1 : numpy.matrix
Vector of known position 1 (CubeSat 1) in meters.
s2 : numpy.matrix
Vector of known position 2 (CubeSat 2) in meters.
k : tuple
Real measurement vector (without errors). It contains the
TDOA in range and two AOA pairs.
Q : numpy.matrix
Measurement noise covariance matrix.
Returns
-------
MSE
The Mean-Square Error.
"""
b1 = self.get_b(k[1], k[2])
b2 = self.get_b(k[3], k[4])
G1 = self.get_Gm(k[1], k[2])
G2 = self.get_Gm(k[3], k[4])
G = self.get_G(b1, b2, G1, G2)
d1 = linalg.norm(u - s1)
d2 = linalg.norm(u - s2)
L1 = self.get_Lm(k[1], k[2])
L2 = self.get_Lm(k[3], k[4])
T1 = self.get_Tm(d1, k[2])
T2 = self.get_Tm(d2, k[4])
T = self.get_T(d1, d2, b1, b2, L1, L2, T1, T2)
T_inv = T.I
G_transpose = G.transpose()
MSE = ((T_inv*G_transpose).transpose()*Q.I*T_inv*G_transpose).I
return MSE
def get_lm(self, u, sm):
lm = sqrt((u[0,0] - sm[0,0])**2 + (u[1,0] - sm[1,0])**2)
return lm
def get_Dm(self, u, sm, lm, dm):
one_div_lm2 = 1/lm**2
aux = (u[2,0] - sm[2,0])/(dm**2*lm)
Dm = matrix([[-(u[1,0] - sm[1,0])*one_div_lm2, (u[0,0] - sm[0,0])*one_div_lm2, 0],
[-(u[0,0] - sm[0,0])*aux, -(u[1,0] - sm[1,0])*aux, lm/dm**2]])
return Dm
def get_FIM(self, u, s1, s2, Q):
"""
Return the Fisher information matrix.
Parameters
----------
u : numpy.matrix
Vector of the unknown source position (femto-satellite) in
meters.
s1 : numpy.matrix
Vector of known position 1 (CubeSat 1) in meters.
s2 : numpy.matrix
Vector of known position 2 (CubeSat 2) in meters.
Q : numpy.matrix
Measurement noise covariance matrix.
Returns
-------
FIM
The Fisher Information Matrix.
"""
d1 = linalg.norm(u - s1)
d2 = linalg.norm(u - s2)
c = (u - s2)/d2 - (u - s1)/d1
l1 = self.get_lm(u, s1)
l2 = self.get_lm(u, s2)
D1 = self.get_Dm(u, s1, l1, d1)
D2 = self.get_Dm(u, s2, l2, d2)
dk_duT = matrix([[c[0,0], c[1,0], c[2,0]],
[D1[0,0], D1[0,1], D1[0,2]],
[D1[1,0], D1[1,1], D1[1,2]],
[D2[0,0], D2[0,1], D2[0,2]],
[D2[1,0], D2[1,1], D2[1,2]]])
FIM = dk_duT.transpose()*Q.I*dk_duT
return FIM
def estimate(self, s1, s2, k, e, Q):
"""
Estimates the unknown source position (femto-satellite)
using the known positions (CubeSats 1 and 2) with their
measurements k with errors e and the measurement noise
covariance matrix Q. This is done according to the method
described in the research paper called: A simple and
accurate TDOA-AOA localization method using two stations.
Parameters
----------
s1 : numpy.matrix
Vector of known position 1 (CubeSat 1) in meters.
s2 : numpy.matrix
Vector of known position 2 (CubeSat 2) in meters.
k : tuple
Real measurement vector (without errors). It contains the
TDOA in range and two AOA pairs.
e : numpy.ndarray
Error vector of the TDOA in range and two AOA pairs.
Q : numpy.matrix
Measurement noise covariance matrix.
Returns
-------
u
Vector of the estimation of the unknown source position
(femto-satellite) in meters.
"""
d21_hat = k[0] + e[0]
theta1_hat = k[1] + e[1]
phi1_hat = k[2] + e[2]
theta2_hat = k[3] + e[3]
phi2_hat = k[4] + e[4]
b1_hat = self.get_b(theta1_hat, phi1_hat)
b2_hat = self.get_b(theta2_hat, phi2_hat)
G1_hat = self.get_Gm(theta1_hat, phi1_hat)
G2_hat = self.get_Gm(theta2_hat, phi2_hat)
G_hat = self.get_G(b1_hat, b2_hat, G1_hat, G2_hat)
G_hat_transpose = G_hat.transpose()
h_hat = self.get_h(d21_hat, s1, s2, b1_hat, b2_hat, G1_hat, G2_hat)
initial_estimate = (G_hat*G_hat_transpose).I*G_hat*h_hat
for i in range(2):
d1 = linalg.norm(initial_estimate - s1)
d2 = linalg.norm(initial_estimate - s2)
L1 = self.get_Lm(theta1_hat, phi1_hat)
L2 = self.get_Lm(theta2_hat, phi2_hat)
T1 = self.get_Tm(d1, phi1_hat)
T2 = self.get_Tm(d2, phi2_hat)
T = self.get_T(d1, d2, b1_hat, b2_hat, L1, L2, T1, T2)
W = self.get_W(Q, T)
if (linalg.det(W) == 0):
return initial_estimate
else:
W_inv = W.I
u_hat = (G_hat*W_inv*G_hat_transpose).I*G_hat*W_inv*h_hat
initial_estimate = u_hat
return u_hat
def RMSE(self, u, estimations, L):
"""
Returns the root-mean-square error.
Parameters
----------
u : numpy.matrix
Vector of the unknown source (femto-satellite)
real position in meters.
estimations : list
L estimations of the unknown source position.
L : int
Number of ensemble runs.
Returns
-------
RMSE
The Root-Mean-Square Error.
"""
aux = 0
for est in estimations:
aux += (est[0,0] - u[0,0])**2 + (est[1,0] - u[1,0])**2 + (est[2,0] - u[2,0])**2
return aux/L
def Bias(self, u, estimations, L):
"""
Returns the estimation bias.
Parameters
----------
u : numpy.matrix
Vector of the unknown source (femto-satellite)
real position in meters.
estimations : list
L estimations of the unknown source position.
L : int
Number of ensemble runs.
Returns
-------
bias
Estimation bias.
"""
acum_u_hat = matrix([[0.0], [0.0], [0.0]])
for est in estimations:
acum_u_hat += est
mean_u_hat = acum_u_hat/L
bias = (mean_u_hat[0,0] - u[0,0])**2 + (mean_u_hat[1,0] - u[1,0])**2 + (mean_u_hat[2,0] - u[2,0])**2
return bias
def get_GNSS_noise(self, std_GNSS, L):
"""
Creates an L length vector of zero mean Gaussian noise.
It considers the standard deviation of a GNSS device.
Parameters
----------
std_GNSS : float
Standard deviation of the GNSS device in meters.
L : int
Number of ensemble runs.
Returns
-------
e
Zero mean Gaussian noise to model the Error of the
two known positions s1 and s2 considering a GNSS
device with a standard deviation std_GNSS.
"""
accuracy_by_axis = std_GNSS/sqrt(3)
e = random.randn(6, L)
e[0,:] = accuracy_by_axis*(e[0,:] - mean(e[0,:]))
e[1,:] = accuracy_by_axis*(e[1,:] - mean(e[1,:]))
e[2,:] = accuracy_by_axis*(e[2,:] - mean(e[2,:]))
e[3,:] = accuracy_by_axis*(e[3,:] - mean(e[3,:]))
e[4,:] = accuracy_by_axis*(e[4,:] - mean(e[4,:]))
e[5,:] = accuracy_by_axis*(e[5,:] - mean(e[5,:]))
return e
def add_GNSS_error(self, s1, s2, e):
"""
Adds noise to the reference positions s1 and s2 considering
the accuracy of the GNSS device.
Parameters
----------
s1 : numpy.matrix
Vector of known position 1 (CubeSat 1) in meters.
s2 : numpy.matrix
Vector of known position 2 (CubeSat 2) in meters.
e : numpy.ndarray
Vector that models the noise of the GNSS devices
for both s1 and s2 known positions.
Returns
-------
noisy_s1
Position vector 1 (CubeSat 1) with zero mean Gaussian
noise due to the GNSS device accuracy.
noisy_s2
Position vector 2 (CubeSat 2) with zero mean Gaussian
noise due to the GNSS device accuracy.
"""
noisy_s1 = matrix([[s1[0,0] + e[0]],
[s1[1,0] + e[1]],
[s1[2,0] + e[2]]])
noisy_s2 = matrix([[s2[0,0] + e[3]],
[s2[1,0] + e[4]],
[s2[2,0] + e[5]]])
return noisy_s1, noisy_s2
def get_deployment_noise(self, std_ADS, std_ACS, N):
"""
Creates an N length vector of zero mean Gaussian noise.
It considers the standard deviation of the attitude
determination and control system of the CubeSat that
deploys the femto-satellite.
Parameters
----------
std_ADS : float
Standard deviation of the satellite's attitude
determination system in degrees.
std_ACS : float
Standard deviation of the satellite's attitude
control system in degrees.
N : int
Number of deployments to be simulated.
Returns
-------
e
Deployment direction noise given that both the attitude
determination and the attitude control systems are not
perfect.
"""
deg2rad = pi/180
e = random.randn(2, N)
e[0,:] = sqrt(std_ADS**2 + std_ACS**2)*deg2rad*(e[0,:] - mean(e[0,:]))
e[1,:] = sqrt(std_ADS**2 + std_ACS**2)*deg2rad*(e[1,:] - mean(e[1,:]))
return e
def noisy_dep_velocity(self, v, e):
"""
Adds noise to the ideal deployment velocity to simulate
the error of the attitude determination and control system
(ADCS) of the CubeSat that deploys the femto-satellite.
Parameters
----------
v : list
Intended deployment velocity (without ADCS noise).
e : numpy.ndarray
Error of the ADCS.
Returns
-------
new_vel
Actual deployent velocity considering the accuracy
of the ADCS.
"""
yaw_noise = e[0]
pitch_noise = e[1]
yaw = arctan2(v[1], v[0]) + yaw_noise
pitch = arctan2(v[2], sqrt(v[0]**2 + v[1]**2)) + pitch_noise
radius = sqrt(v[0]**2 + v[1]**2 + v[2]**2)
new_vel = [radius*cos(yaw)*cos(pitch), radius*sin(yaw)*cos(pitch), radius*sin(pitch)]
return new_vel
def simulation2(self, s1, s2, L, std_RD):
rmse = zeros(len(std_RD))
bias = zeros(len(std_RD))
rcrb = zeros(len(std_RD))
u = matrix([[1000.0], [1000.0], [1000.0]])
k = self.get_real_vector(u, s1, s2)
std_AOA = 0.5*pi/180.0
for i, sigma in enumerate(std_RD):
estimations = []
Q = self.get_Q(sigma, std_AOA)
MSE = self.get_MSE(u, s1, s2, k, Q)
e = self.get_error_vector(sigma, std_AOA, L)
for j in range(L):
u_hat = self.estimate(s1, s2, k, e[:,j], Q)
estimations.append(u_hat)
rmse[i] = self.RMSE(u, estimations, L)
bias[i] = self.Bias(u, estimations, L)
rcrb[i] = MSE[0,0] + MSE[1,1] + MSE[2,2]
return rmse, bias, rcrb
def plot_fig2(self, L):
"""
Figure 2 of A simple and accurate TDOA-AOA localization
method using two stations.
Parameters
----------
L : int
Number of ensemble runs.
"""
rmse = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
bias = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
rcrb = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
std_RD = [10**0.5, 10, 10**1.5, 10**2, 10**2.5, 10**3]
rmse = zeros(len(std_RD))
bias = zeros(len(std_RD))
rcrb = zeros(len(std_RD))
radius = 300.0
deg2rad = pi/180.0
L0 = 10
for i in range(L0):
theta_s = random.uniform(-179, 179)*deg2rad
phi_s = random.uniform(-89, 89)*deg2rad
#s1 = radius*matrix([[cos(theta_s)*cos(phi_s)],
# [sin(theta_s)*cos(phi_s)],
# [sin(phi_s)]])
s1 = radius*matrix([[cos(theta_s)], [sin(theta_s)], [0]])
s2 = -s1
rm, b, rc = self.simulation2(s1, s2, L, std_RD)
rmse += rm
bias += b
rcrb += rc
rmse = sqrt(rmse/L0)
bias = sqrt(bias/L0)
rcrb = sqrt(rcrb/L0)
fig, ax = subplots(1, 1, sharey=False)
ax.loglog(std_RD, rmse, 'o', linewidth=2.0, markersize=12, clip_on=False,
fillstyle="none", label="RMSE of the proposed method")
ax.loglog(std_RD, rcrb, '-', linewidth=2.0, markersize=12,
label="Root CRB")
ax.loglog(std_RD, bias, '+', linewidth=2.0, markersize=12, clip_on=False,
label="Bias of the proposed method")
ax.grid()
ax.legend(fontsize=14, loc="center right")
ax.set(xlabel="{} [m]".format(r'$\sigma_{RD}$'), ylabel="RMSE and bias [m]")
ax.xaxis.label.set_size(16)
ax.yaxis.label.set_size(16)
ax.tick_params(which="both", direction="in", labelsize=14,
bottom=True, top=True, left=True, right=True)
axis((10**0.5, 10**3, 10**(-1), 10**2))
tight_layout()
def simulation3(self, s1, s2, L, std_AOA):
rmse = zeros(len(std_AOA))
bias = zeros(len(std_AOA))
rcrb = zeros(len(std_AOA))
u = matrix([[1000.0], [1000.0], [1000.0]])
k = self.get_real_vector(u, s1, s2)
std_RD = 10.0
for i, sigma in enumerate(std_AOA):
estimations = []
Q = self.get_Q(std_RD, sigma)
MSE = self.get_MSE(u, s1, s2, k, Q)
e = self.get_error_vector(std_RD, sigma, L)
for j in range(L):
u_hat = self.estimate(s1, s2, k, e[:,j], Q)
estimations.append(u_hat)
rmse[i] = self.RMSE(u, estimations, L)
bias[i] = self.Bias(u, estimations, L)
rcrb[i] = MSE[0,0] + MSE[1,1] + MSE[2,2]
return rmse, bias, rcrb
def plot_fig3(self, L):
"""
Figure 3 of A simple and accurate TDOA-AOA localization
method using two stations.
Parameters
----------
L : int
Number of ensemble runs.
"""
std_AOA_deg = [0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5]
deg2rad = pi/180.0
std_AOA = [i*deg2rad for i in std_AOA_deg]
rmse = zeros(len(std_AOA))
bias = zeros(len(std_AOA))
rcrb = zeros(len(std_AOA))
radius = 300.0
L0 = 10
for i in range(L0):
theta_s = random.uniform(-179, 179)*deg2rad
phi_s = random.uniform(-89, 89)*deg2rad
#s1 = radius*matrix([[cos(theta_s)*cos(phi_s)],
# [sin(theta_s)*cos(phi_s)],
# [sin(phi_s)]])
s1 = radius*matrix([[cos(theta_s)], [sin(theta_s)], [0]])
s2 = -s1
rm, b, rc = self.simulation3(s1, s2, L, std_AOA)
rmse += rm
bias += b
rcrb += rc
rmse = sqrt(rmse/L0)
bias = sqrt(bias/L0)
rcrb = sqrt(rcrb/L0)
fig, ax = subplots(1, 1, sharey=False)
ax.semilogy(std_AOA_deg, rmse, 'o', linewidth=2.0, markersize=12, clip_on=False,
fillstyle="none", label="RMSE of the proposed method")
ax.semilogy(std_AOA_deg, rcrb, '-', linewidth=2.0, markersize=12,
label="Root CRB")
ax.semilogy(std_AOA_deg, bias, '+', linewidth=2.0, markersize=12, clip_on=False,
label="Bias of the proposed method")
ax.grid()
ax.legend(fontsize=14, loc="center right")
ax.set(xlabel="{} [deg]".format(r'$\sigma_{AOA}$'), ylabel="RMSE and bias [m]")
ax.xaxis.label.set_size(16)
ax.yaxis.label.set_size(16)
ax.tick_params(which="both", direction="in", labelsize=14,
bottom=True, top=True, left=True, right=True)
axis((0.5, 2.5, 10**(-1), 10**3))
xticks([0.5, 1, 1.5, 2, 2.5])
tight_layout()
def simulation4(self, s1, s2, L, ratio):
rmse = zeros(len(ratio))
bias = zeros(len(ratio))
rcrb = zeros(len(ratio))
std_RD = 10.0
std_AOA = 1.0*pi/180.0
Q = self.get_Q(std_RD, std_AOA)
base_u = sqrt(3)*matrix([[100.0], [100.0], [100.0]])
for i, a in enumerate(ratio):
estimations = []
u = a*base_u
k = self.get_real_vector(u, s1, s2)
MSE = self.get_MSE(u, s1, s2, k, Q)
e = self.get_error_vector(std_RD, std_AOA, L)
for j in range(L):
u_hat = self.estimate(s1, s2, k, e[:,j], Q)
estimations.append(u_hat)
rmse[i] = self.RMSE(u, estimations, L)
bias[i] = self.Bias(u, estimations, L)
rcrb[i] = MSE[0,0] + MSE[1,1] + MSE[2,2]
return rmse, bias, rcrb
def plot_fig4(self, L):
"""
Figure 4 of A simple and accurate TDOA-AOA localization
method using two stations.
Parameters
----------
L : int
Number of ensemble runs.
"""
ratio = [2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0]
rmse = zeros(len(ratio))
bias = zeros(len(ratio))
rcrb = zeros(len(ratio))
radius = 300.0
deg2rad = pi/180.0
L0 = 10
for i in range(L0):
theta_s = random.uniform(-179, 179)*deg2rad
phi_s = random.uniform(-89, 89)*deg2rad
#s1 = radius*matrix([[cos(theta_s)*cos(phi_s)],
# [sin(theta_s)*cos(phi_s)],
# [sin(phi_s)]])
s1 = radius*matrix([[cos(theta_s)], [sin(theta_s)], [0]])
s2 = -s1
rm, b, rc = self.simulation4(s1, s2, L, ratio)
rmse += rm
bias += b
rcrb += rc
rmse = sqrt(rmse/L0)
bias = sqrt(bias/L0)
rcrb = sqrt(rcrb/L0)
fig, ax = subplots(1, 1, sharey=False)
ax.semilogy(ratio, rmse, 'o', linewidth=2.0, markersize=12, clip_on=False,
fillstyle="none", label="RMSE of the proposed method")
ax.semilogy(ratio, rcrb, '-', linewidth=2.0, markersize=12,
label="Root CRB")
ax.semilogy(ratio, bias, '+', linewidth=2.0, markersize=12, clip_on=False,
label="Bias of the proposed method")
ax.grid()
ax.legend(fontsize=14, loc="center right")
ax.set(xlabel="Source-to-station-range ratio a", ylabel="RMSE and bias [m]")
ax.xaxis.label.set_size(16)
ax.yaxis.label.set_size(16)
ax.tick_params(which="both", direction="in", labelsize=14,
bottom=True, top=True, left=True, right=True)
axis((2, 10, 10**(-1.2), 10**3))
xticks([2, 4, 6, 8, 10])
tight_layout()
def simulation_GNSS(self, sat_u, sat_s1, sat_s2, L, std_GNSS, date):
"""
Runs L estimations for each standard deviation of the GNSS
devices contained in std_GNSS. These standard deviations are
for the GNSS devices used by the CubeSats 1 and 2.
Parameters
----------
sat_u : pypredict.sat.Sat
Satellite object that describes all the orbital
parameters and position of the femto-satellite.
sat_s1 : pypredict.sat.Sat
Satellite object that describes all the orbital
parameters and position of the CubeSat 1.
sat_s2 : pypredict.sat.Sat
Satellite object that describes all the orbital
parameters and position of the CubeSat 2.
L : int
Number of ensemble runs.
std_GNSS : numpy.ndarray
Array of standard deviations of the GNSS devices
in meters.
date : datetime.datetime
Simulation date.
Returns
-------
rmse
The Root-Mean-Square Error.
bias
Estimation bias.
rcrb
The Root Cramer-Rao Bound.
"""
rmse = zeros(len(std_GNSS))
bias = zeros(len(std_GNSS))
rcrb = zeros(len(std_GNSS))
std_RD = 10.0
std_AOA = 1.0*pi/180.0
Q = self.get_Q(std_RD, std_AOA)
sat_u.updateOrbitalParameters(date)
sat_s1.updateOrbitalParameters(date)
sat_s2.updateOrbitalParameters(date + timedelta(seconds=4))
u = sat_u.getXYZ()
s1 = sat_s1.getXYZ()
s2 = sat_s2.getXYZ()
k = self.get_real_vector(u, s1, s2)
MSE = self.get_MSE(u, s1, s2, k, Q)
for i, std in enumerate(std_GNSS):
estimations = []
e = self.get_error_vector(std_RD, std_AOA, L)
GNSS_noise = self.get_GNSS_noise(std, L)
for j in range(L):
noisy_s1, noisy_s2 = self.add_GNSS_error(s1, s2, GNSS_noise[:,j])
k_w_GNSS_error = self.get_real_vector(u, noisy_s1, noisy_s2)
u_hat = self.estimate(noisy_s1, noisy_s2, k_w_GNSS_error, e[:,j], Q)
estimations.append(u_hat)
rmse[i] = self.RMSE(u, estimations, L)
bias[i] = self.Bias(u, estimations, L)
rcrb[i] = MSE[0,0] + MSE[1,1] + MSE[2,2]
print(sqrt(rcrb[i]))
return rmse, bias, rcrb
def plot_fig_GNSS(self, L, dep_date, output="rcrb_GNSS.csv"):
"""
This method studies the effect of the accuracy of the GNSS
devices used by the CubeSats in the femto-satellite's
position estimation. It simulates L0 deployments, and for
each deployment, L ensemble runs for each GNSS standard
deviation.
This method generates Figure 7 of the research paper called:
A femto-satellite localization method based on TDOA and AOA
using two Cubesats.
Parameters
----------
L : int
Number of ensemble runs.
dep_date : datetime.datetime
Date on which the deployment takes place.
"""
start = datetime.utcnow()
std_GNSS = array([10.0, 50.0, 100.0, 120.0])
std_ADS = 36/3600 # STT with 36 arcseconds of accuracy.
std_ACS = 0.06 # RWs of 0.06° of accuracy.
rmse = zeros(len(std_GNSS))
bias = zeros(len(std_GNSS))
rcrb = zeros(len(std_GNSS))
dpl = Dpl()
s1_mass = 3.2
u_mass = 0.08
L0 = 5000