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python_exercise_2_13.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
MIT License
Copyright (c) 2022 Jongrae.K
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.
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp
def angular_velocity_true(time_c):
w_true = np.array([ 0.01*np.sin(2*np.pi*0.005*time_c), # [rad/s]
0.05*np.cos(2*np.pi*0.001*time_c + 0.2), #[rad/s]
0.02 #[rad/s]
])
return w_true
def q2dcm(quat):
quat = quat.squeeze()
q13 =quat[0:3]
q4 = quat[3]
q13x = np.array([[ 0, -q13[2], q13[1]],
[q13[2], 0, -q13[0]],
[-q13[1], q13[0], 0]])
q13 = q13.reshape(3,1)
dcm = (q4**2-q13.T@q13)*np.eye(3) + 2*([email protected]) - 2*q4*q13x
return dcm
def dqdt_attitude_kinematics(time, state):
quat = state
w_true = angular_velocity_true(time) #[rad/s]
wx=np.array([[0, -w_true[2], w_true[1]],
[w_true[2], 0, -w_true[0]],
[-w_true[1], w_true[0], 0]])
Omega_13 = np.hstack((-wx,np.resize(w_true,(3,1))))
Omega_4 = np.hstack((-w_true,0))
Omega = np.vstack((Omega_13, Omega_4))
dqdt = 0.5*(Omega@quat)
return dqdt
def kalman_filter_attitude(x_hat_0, P0, dt_KF,
rR_star_all, rB_star_measure, w_measure,
sgm_v, sgm_u, sgm_star):
num_KF_state = 6 # [dq1 dq2 dq3 b1 b2 b3]
q_est = x_hat_0[0:4]; q_est = q_est.reshape(4,1)
b_est = x_hat_0[4:]; b_est = b_est.reshape(3,1)
w_hat = w_measure.reshape(3,1) - b_est
w_hat_mag = np.sqrt(w_hat.T@w_hat)[0,0]
w_hatx = np.array([[0, -w_hat[2,0], w_hat[1,0]],
[w_hat[2,0], 0, -w_hat[0,0]],
[-w_hat[1,0], w_hat[0,0], 0]])
# propagate
if w_hat_mag > 1e-12:
dtheta_k = w_hat_mag*dt_KF
cos_th = np.cos(dtheta_k/2)
sin_th_over_w = np.sin(dtheta_k/2)/w_hat_mag
q_Phi_1st_row = np.hstack((cos_th*np.eye(3)-sin_th_over_w*w_hatx,sin_th_over_w*w_hat))
q_Phi_2nd_row = np.hstack((-sin_th_over_w*w_hat.T[0],cos_th))
q_Phi = np.vstack((q_Phi_1st_row,q_Phi_2nd_row))
cos_th = np.cos(w_hat_mag*dt_KF)
sin_th = np.sin(w_hat_mag*dt_KF)
Phi_1 = np.eye(3) - w_hatx*sin_th/w_hat_mag + (w_hatx@w_hatx)*((1-cos_th)/w_hat_mag**2)
Phi_2 = -np.eye(3)*dt_KF + w_hatx*((1-cos_th)/w_hat_mag**2) - (w_hatx@w_hatx)*((w_hat_mag*dt_KF-sin_th)/w_hat_mag**3)
else:
q_Phi = np.vstack((np.eye(3)-(dt_KF/2)*w_hatx,-(dt_KF/2)*w_hat.T))
Phi_1 = np.eye(3) - w_hatx*dt_KF
Phi_2 = -np.eye(3)*dt_KF
q_est_minus = q_Phi@q_est; q_est_minus = q_est_minus/np.linalg.norm(q_est_minus)
b_est_minus = b_est
dcm_BR_minus = q2dcm(q_est_minus)
Q_1st_row = np.hstack(((sgm_v**2*dt_KF+(dt_KF**3/3)*sgm_u**2)*np.eye(3),-(dt_KF**2/2)*sgm_u**2*np.eye(3)-(dt_KF**3/6)*sgm_u**2*w_hatx))
Q_2nd_row = np.hstack((-(dt_KF**2/2)*sgm_u**2*np.eye(3)-(dt_KF**3/6)*sgm_u**2*w_hatx,sgm_u**2*dt_KF*np.eye(3)))
Q = np.vstack((Q_1st_row,Q_2nd_row))
Phi_1st_row = np.hstack((Phi_1,Phi_2))
Phi_2nd_row = np.hstack((np.zeros((3,3)),np.eye(3)))
Phi = np.vstack((Phi_1st_row,Phi_2nd_row))
P1 = Phi@[email protected] + Q
# update
num_star = rB_star_measure.shape[1]
rB_star_hat = dcm_BR_minus@rR_star_all
H_k = np.zeros((3*num_star,6))
R = sgm_star**2*np.eye(num_star*3)
for xdx in range(num_star):
vec = rB_star_hat[:,xdx]
vec_x = np.array([[0, -vec[2], vec[1]],
[vec[2], 0, -vec[0]],
[-vec[1], vec[0], 0]])
st_idx = 3*xdx
H_k[st_idx:st_idx+3,:] = np.hstack((vec_x,np.zeros((3,3))))
K_k = P1@[email protected](H_k@P1@H_k.T+R)
P1 = (np.eye(num_KF_state)-K_k@H_k)@P1
rB_star_mea_vec = rB_star_measure.T.reshape(3*num_star,1)
rB_star_hat_vec = rB_star_hat.T.reshape(3*num_star,1)
delta_x = K_k@(rB_star_mea_vec-rB_star_hat_vec)
# quaternion & bias update
dq_13 = 2*delta_x[0:3]; dq_13 = dq_13.reshape((3,1))
q = q_est_minus.squeeze()
qx = np.array([[0, -q[2], q[1]],
[q[2],0,-q[0]],
[-q[1],q[0],0]])
quat_update_matrix = np.vstack((q[3]*np.eye(3)+qx,-q[0:3].T))
q_hat_plus = q_est_minus + quat_update_matrix@dq_13; q_hat_plus = q_hat_plus/np.linalg.norm(q_hat_plus)
b_hat_plus = b_est_minus + delta_x[3:]
x_hat_1 = np.vstack((q_hat_plus,b_hat_plus))
return x_hat_1, P1
#----------------------------------------------------------------------------
# Set initial values & change non-SI units into the SI Units
dt = 0.05 # [seconds]
time_init = 0
time_final = 120 # [seconds]
N_sample = int(time_final/dt) + 1
time = np.linspace(time_init,time_final, N_sample)
# standard deviation of the bias, sigma_beta_xyz
sigma_beta = 0.0005 # [degrees/sqrt(s)]
sigma_u = sigma_beta*(np.pi/180) # [rad/sqrt(s)]
sigma_eta = sigma_u/np.sqrt(dt)
# standard devitation of the white noise, sigma_v
sigma_v = 0.0001 #[degrees/s]
sigma_v = sigma_v*(np.pi/180) #[rad/s]
# initial beta(t)
beta = (2*np.random.rand(3)-1)*0.03 # +/- 0.03[degrees/s]
beta = beta*(np.pi/180) # [radians/s]
# prepare the data store
w_all = np.zeros((N_sample,3))
w_measure_all = np.zeros((N_sample,3))
# data store
# instead of calculating the exact size
# of the following matrices, use varying matrices with increasing
# time, which might not be significant but simpler to implement
w_gyr_all = []
w_hat_all = []
w_tr_all = []
q_tr_all = []
q_hat_all = []
time_all = []
pcov_all = []
q_current = np.array([0,0,0,1])
# star sensor
# star sensor reference star vectors
r1R = np.array([-0.6794, -0.3237, -0.6586]).reshape((3,1))
r2R = np.array([-0.7296, 0.5858, 0.3528]).reshape((3,1))
r3R = np.array([-0.2718, 0.6690, -0.6918]).reshape((3,1))
r4R = np.array([-0.2062, -0.3986, 0.8936]).reshape((3,1))
r5R = np.array([0.6858, -0.7274, -0.0238]).reshape((3,1))
r1R = r1R/np.sqrt(r1R.T@r1R)[0,0]
r2R = r2R/np.sqrt(r2R.T@r2R)[0,0]
r3R = r3R/np.sqrt(r3R.T@r3R)[0,0]
r4R = r4R/np.sqrt(r4R.T@r4R)[0,0]
r5R = r5R/np.sqrt(r5R.T@r5R)[0,0]
rR_star_all = np.hstack((r1R,r2R,r3R,r4R,r5R))
num_star = rR_star_all.shape[1]
sigma_star = 87.2665/3*1e-6
r_star = sigma_star**2*np.eye(num_star*3)
# Kalman filter
n_dt_KF = 2
dt_KF = n_dt_KF*dt
bias_estimate_current = np.zeros((3,1))
q_estimate_current = np.array([0, 0, 0, 1]).reshape((4,1)) + 0.0*np.random.randn(4,1)
q_estimate_current = q_estimate_current/np.linalg.norm(q_estimate_current)
x0 = np.vstack((q_estimate_current, bias_estimate_current))
p_current = 0.001*np.eye(6)
w_hat = np.array([0, 0, 0]).reshape((3,1))
# main simulation loops
for idx in range(N_sample):
time_c = time[idx]
w_true = angular_velocity_true(time_c)
# beta(t)
eta_u = sigma_eta*np.random.randn(3)
dbeta = eta_u*dt
beta = beta + dbeta
# eta_v(t)
eta_v = sigma_v*np.random.randn(3)
# w_tilde
w_measurement = w_true + beta + eta_v
if np.remainder(idx,n_dt_KF)==1:
# star sensor measurement
dcm_BR = q2dcm(q_current)
rB_star_all = dcm_BR@rR_star_all
rB_star_measure = rB_star_all+sigma_star*np.random.randn(3,num_star)
rB_star_measure = rB_star_measure/np.kron(np.ones((3,1)),np.sqrt(np.sum(rB_star_measure**2,0)))
# kalman filter
x_hat_1, P1 = kalman_filter_attitude(x0, p_current, dt_KF,
rR_star_all, rB_star_measure, w_measurement,
sigma_v, sigma_u, sigma_star)
x0 = x_hat_1
p_current = P1
q_estimate_current = x0[0:4]
q_estimate_current = q_estimate_current/np.linalg.norm(q_estimate_current)
bias_estimate_current = x0[4:]
w_hat = w_measurement.reshape((3,1)) - bias_estimate_current
# store data to plot
# instead of calculating the exact size
# of the following matrices, use varying matrices with increasing
# time, which might not be significant but simpler to implement
time_all.append(time_c)
w_gyr_all.append(w_measurement.squeeze())
w_hat_all.append(w_hat.squeeze())
w_tr_all.append(w_true.squeeze())
q_tr_all.append(q_current.squeeze())
q_hat_all.append(q_estimate_current.squeeze())
pcov_all.append(np.diag(P1).squeeze())
# integrate true dqdt to obtain true q(t): time_c -> time_c + dt
if idx < N_sample-1:
sol = solve_ivp(dqdt_attitude_kinematics, (time_c, time[idx+1]), q_current)
q_current = sol.y[:,-1]
# make the lists to numpy arrays
q_tr_all=np.array(q_tr_all)
q_hat_all=np.array(q_hat_all)
dq = q_tr_all-q_hat_all
w_tr_all=np.array(w_tr_all)
w_hat_all=np.array(w_hat_all)
dw = w_tr_all - w_hat_all
pcov_all = np.array(pcov_all)
# plot results
fig0, (ax0,ax1,ax2) = plt.subplots(nrows=3,ncols=1)
ax0.plot(time_all,dq[:,0],time_all,-3*np.sqrt(pcov_all[:,0]),'r--',time_all,3*np.sqrt(pcov_all[:,0]),'r--')
ax0.axis((time_init,time_final,-4e-5,4e-5))
ax0.set_ylabel('$\delta q_1$',fontsize=14)
ax0.legend(('error','3$\sigma$ bound'),fontsize=8, loc='upper right')
ax1.plot(time_all,dq[:,1],time_all,-3*np.sqrt(pcov_all[:,1]),'r--',time_all,3*np.sqrt(pcov_all[:,1]),'r--')
ax1.axis((time_init,time_final,-4e-5,4e-5))
ax1.set_ylabel('$\delta q_2$',fontsize=14)
ax1.legend(('error','3$\sigma$ bound'),fontsize=8, loc='upper right')
ax2.plot(time_all,dq[:,2],time_all,-3*np.sqrt(pcov_all[:,2]),'r--',time_all,3*np.sqrt(pcov_all[:,2]),'r--')
ax2.axis((time_init,time_final,-4e-5,4e-5))
ax2.set_ylabel('$\delta q_3$',fontsize=14)
ax2.legend(('error','3$\sigma$ bound'),fontsize=8, loc='upper right')
ax2.set_xlabel('time [s]')
fig1, (bx0,bx1,bx2) = plt.subplots(nrows=3,ncols=1)
bx0.plot(time_all,dw[:,0],time_all,-3*np.sqrt(pcov_all[:,3]),'r--',time_all,3*np.sqrt(pcov_all[:,3]),'r--')
bx0.axis((time_init,time_final,-4e-5,4e-5))
bx0.set_ylabel('$\delta\omega_1$ [rad/s]',fontsize=14)
bx0.legend(('error','3$\sigma$ bound'),fontsize=8, loc='upper right')
bx1.plot(time_all,dw[:,1],time_all,-3*np.sqrt(pcov_all[:,4]),'r--',time_all,3*np.sqrt(pcov_all[:,4]),'r--')
bx1.axis((time_init,time_final,-4e-5,4e-5))
bx1.set_ylabel('$\delta\omega_2$ [rad/s]',fontsize=14)
bx1.legend(('error','3$\sigma$ bound'),fontsize=8, loc='upper right')
bx2.plot(time_all,dw[:,2],time_all,-3*np.sqrt(pcov_all[:,5]),'r--',time_all,3*np.sqrt(pcov_all[:,5]),'r--')
bx2.axis((time_init,time_final,-4e-5,4e-5))
bx2.set_ylabel('$\delta\omega_3$ [rad/s]',fontsize=14)
bx2.legend(('error','3$\sigma$ bound'),fontsize=8, loc='upper right')
bx2.set_xlabel('time [s]')