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ekf_non_lin_helical.py
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ekf_non_lin_helical.py
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import plotly.graph_objects as go
import numpy as np
from math import cos, sin, sqrt
## For modelling the noise that gets added after each update step
def noise(mean_matrix, covariance_matrix):
return np.random.multivariate_normal(mean_matrix, covariance_matrix).reshape(-1, 1)
def action_upd(X_t, A_t, B_t, u_t, mean_epsilon, R):
return np.dot(A_t, X_t) + np.dot(B_t, u_t) + noise(mean_epsilon, R)
def obsv_upd(X_t, C_t, mean_delta, Q):
return np.dot(C_t, X_t) + noise(mean_delta, Q)
## Kalman Filter Update Equations
def kalman_update(mu_tm1, sigma_tm1, u_t, z_t, A_t, B_t, C_t, Q_t, R_t):
n_shape = mu_tm1.shape[0]
C_t_transpose = np.transpose(C_t)
A_t_transpose = np.transpose(A_t)
mu_t_bar = np.dot(A_t, mu_tm1) + np.dot(B_t, u_t)
sigma_t_bar = np.dot(np.dot(A_t, sigma_tm1), A_t_transpose) + R_t
K_t = np.dot(np.dot(sigma_t_bar, C_t_transpose),np.linalg.inv(np.dot(np.dot(C_t, sigma_t_bar),C_t_transpose)+Q_t))
mu_t = mu_t_bar + np.dot(K_t, (z_t-np.dot(C_t, mu_t_bar)))
sigma_t = np.dot((np.identity(n_shape)-np.dot(K_t, C_t)), sigma_t_bar)
return mu_t, sigma_t
## Given the actual belief, compute the measurement the estimator will recieve
def obsv_upd_land(X_t, mean_delta, Q, landmark_num):
x = X_t[0][0]
y = X_t[1][0]
z = X_t[1][0]
if(landmark_num == 1):
h_mu_t = np.array([[x], [y], [z], [sqrt((x-150.0)*(x-150.0) + y*y + (z-100.0)*(z-100.0))]])
elif(landmark_num == 2):
h_mu_t = np.array([[x], [y], [z], [sqrt((x+150.0)*(x+150.0) + y*y + (z-100.0)*(z-100.0))]])
elif(landmark_num == 3):
h_mu_t = np.array([[x], [y], [z], [sqrt(x*x + (y-150.0)*(y-150.0) + (z-100.0)*(z-100.0))]])
elif(landmark_num == 4):
h_mu_t = np.array([[x], [y], [z], [sqrt(x*x + (y+150.0)*(y+150.0) + (z-100.0)*(z-100.0))]])
return h_mu_t + noise(mean_delta, Q)
## Computing the jacobians for linearisation
def jacobian_h_l1(mu):
x = mu[0][0]
y = mu[1][0]
z = mu[2][0]
v1 = (x - 150.0)/(sqrt((x-150.0)*(x-150.0) + y*y + (z-100.0)*(z-100.0)))
v2 = y/(sqrt((x-150.0)*(x-150.0) + y*y + (z-100.0)*(z-100.0)))
v3 = (z-100.0)/(sqrt((x-150.0)*(x-150.0) + y*y + (z-100.0)*(z-100.0)))
H = np.array([[1.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0, 0.0, 0.0],
[ v1, v2, v3, 0.0, 0.0, 0.0]])
return H
def jacobian_h_l2(mu):
x = mu[0][0]
y = mu[1][0]
z = mu[2][0]
v1 = (x + 150.0)/(sqrt((x+150.0)*(x-150.0) + y*y + (z-100.0)*(z-100.0)))
v2 = y/(sqrt((x+150.0)*(x+150.0) + y*y + (z-100.0)*(z-100.0)))
v3 = (z-100.0)/(sqrt((x+150.0)*(x+150.0) + y*y + (z-100.0)*(z-100.0)))
H = np.array([[1.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0, 0.0, 0.0],
[ v1, v2, v3, 0.0, 0.0, 0.0]])
return H
def jacobian_h_l3(mu):
x = mu[0][0]
y = mu[1][0]
z = mu[2][0]
v1 = (x)/(sqrt(x*x + (y-150.0)*(y-150.0) + (z-100.0)*(z-100.0)))
v2 = (y-150.0)/(sqrt(x*x + (y-150.0)*(y-150.0) + (z-100.0)*(z-100.0)))
v3 = (z-100.0)/(sqrt(x*x + (y-150.0)*(y-150.0) + (z-100.0)*(z-100.0)))
H = np.array([[1.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0, 0.0, 0.0],
[ v1, v2, v3, 0.0, 0.0, 0.0]])
return H
def jacobian_h_l4(mu):
x = mu[0][0]
y = mu[1][0]
z = mu[2][0]
v1 = (x)/(sqrt(x*x + (y+150.0)*(y+150.0) + (z-100.0)*(z-100.0)))
v2 = (y+150.0)/(sqrt(x*x + (y+150.0)*(y+150.0) + (z-100.0)*(z-100.0)))
v3 = (z-100.0)/(sqrt(x*x + (y+150.0)*(y+150.0) + (z-100.0)*(z-100.0)))
H = np.array([[1.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0, 0.0, 0.0],
[ v1, v2, v3, 0.0, 0.0, 0.0]])
return H
## Prapagating belief states handling non-linear measurements via EKF
def ext_kalman_update(mu_tm1, sigma_tm1, u_t, z_t, A_t, B_t, Q_t, R_t, landmark_num):
n_shape = mu_tm1.shape[0]
A_t_transpose = np.transpose(A_t)
mu_t_bar = np.dot(A_t, mu_tm1) + np.dot(B_t, u_t)
sigma_t_bar = np.dot(np.dot(A_t, sigma_tm1), A_t_transpose) + R_t # R_t is the motion model noise
x_mu_t_bar = mu_t_bar[0][0]
y_mu_t_bar = mu_t_bar[1][0]
z_mu_t_bar = mu_t_bar[2][0]
if(landmark_num == 1):
H_t = jacobian_h_l1(mu_t_bar)
h_mu_t = np.array([[x_mu_t_bar], [y_mu_t_bar], [z_mu_t_bar], [sqrt((x_mu_t_bar-150.0)*(x_mu_t_bar-150.0) + y_mu_t_bar*y_mu_t_bar + (z_mu_t_bar-100.0)*(z_mu_t_bar-100.0))]])
elif(landmark_num == 2):
H_t = jacobian_h_l2(mu_t_bar)
h_mu_t = np.array([[x_mu_t_bar], [y_mu_t_bar], [z_mu_t_bar], [sqrt((x_mu_t_bar+150.0)*(x_mu_t_bar+150.0) + y_mu_t_bar*y_mu_t_bar + (z_mu_t_bar-100.0)*(z_mu_t_bar-100.0))]])
elif(landmark_num == 3):
H_t = jacobian_h_l3(mu_t_bar)
h_mu_t = np.array([[x_mu_t_bar], [y_mu_t_bar], [z_mu_t_bar], [sqrt(x_mu_t_bar*x_mu_t_bar + (y_mu_t_bar-150.0)*(y_mu_t_bar-150.0) + (z_mu_t_bar-100.0)*(z_mu_t_bar-100.0))]])
elif(landmark_num == 4):
H_t = jacobian_h_l4(mu_t_bar)
h_mu_t = np.array([[x_mu_t_bar], [y_mu_t_bar], [z_mu_t_bar], [sqrt(x_mu_t_bar*x_mu_t_bar + (y_mu_t_bar+150.0)*(y_mu_t_bar+150.0) + (z_mu_t_bar-100.0)*(z_mu_t_bar-100.0))]])
H_t_transpose = np.transpose(H_t)
K_t = np.dot(np.dot(sigma_t_bar, H_t_transpose),np.linalg.inv(np.dot(np.dot(H_t, sigma_t_bar),H_t_transpose)+Q_t))
mu_t = mu_t_bar + np.dot(K_t, (z_t-h_mu_t))
sigma_t = np.dot((np.identity(n_shape)-np.dot(K_t, H_t)), sigma_t_bar)
return mu_t, sigma_t
def generate_ellipsoid_points(mu_t, sigma_t, num_points=10):
mean = mu_t[0:3].reshape(-1, 1)
std_dev = sigma_t[:3, :3]
u = np.linspace(0, 2 * np.pi, num_points)
v = np.linspace(0, np.pi, num_points)
x = mean[0] + std_dev[0][0] * np.outer(np.cos(u), np.sin(v))
y = mean[1] + std_dev[1][1] * np.outer(np.sin(u), np.sin(v))
z = mean[2] + std_dev[2][2] * np.outer(np.ones(np.size(u)), np.cos(v))
return x.flatten(), y.flatten(), z.flatten()
def uncer_ellipse_params(mu_t, sigma_t):
xyz = mu_t[0:3].reshape(-1, 1)
xyz_cov = sigma_t[:3, :3]
eigen_values, eigen_vectors = np.linalg.eig(xyz_cov)
std_dev_major = np.sqrt(eigen_values[0])
std_dev_minor = np.sqrt(eigen_values[1])
rotation_angle = np.arctan2(eigen_vectors[1, 0], eigen_vectors[0, 0])
theta = np.linspace(0, 2 * np.pi, 100)
x_rotated = mu_t[0] + (std_dev_major * np.cos(theta)) * np.cos(rotation_angle) - (std_dev_minor * np.sin(theta)) * np.sin(rotation_angle)
y_rotated = mu_t[1] + (std_dev_major * np.cos(theta)) * np.sin(rotation_angle) + (std_dev_minor * np.sin(theta)) * np.cos(rotation_angle)
return x_rotated, y_rotated
## Simulate and Plot the estimates
def ekf_for_landmarks_helical():
X_init = np.array([[100.0], [0.0], [0.0], [0.0], [4.0], [0.0]]) ## start from (150,0,0) with vel (0, v, 0)
mu_init = np.array([[100.0], [0.0], [0.0], [0.0], [4.0], [0.0]])
sigma_init = (0.008)*(0.008)*np.identity(6) ## Prior belief covariance
#observed_traj = np.zeros((300, 2))
num_t_steps = 200
actual_traj = np.zeros((num_t_steps, 3))
estimated_traj = np.zeros((num_t_steps, 3))
#uncertainity_ellipse_values = np.zeros((num_t_steps, 100, 2))
L1 = np.array([[ 150.0], [0.0], [100.0]])
L2 = np.array([[-150.0], [0.0], [100.0]])
L3 = np.array([[0.0], [ 150.0], [100.0]])
L4 = np.array([[0.0], [-150.0], [100.0]])
landmark_range = 90.0
uncertainity_ellipse_values = np.zeros((num_t_steps, 100, 3))
print("xt shape = ", X_init.shape)
## Action Model Parameters
deltaT = 1.0
A_t = np.array([[1.0,0,0,deltaT,0,0],
[0,1.0,0,0,deltaT,0],
[0,0,1.0,0,0,deltaT],
[0,0,0,1.0,0,0],
[0,0,0,0,1.0,0],
[0,0,0,0,0,1.0]])
B_t = np.array([[0,0,0],
[0,0,0],
[0,0,0],
[1.0,0,0],
[0,1.0,0],
[0,0,1.0]])
mean_epsilon = np.zeros(6)
R = (0.01)*(0.01)*np.identity(6) ## isotropic Gaussian Noise
## Observation Model Parameters
C_t = np.array([[1.0,0,0,0,0,0],
[0,1.0,0,0,0,0],
[0,0,1.0,0,0,0]])
sigma_gps = 10.0
sigma_lm = 0.001
Q_gps_only = sigma_gps*sigma_gps*np.identity(3)
Q_gps_landmark = np.diag(np.array([sigma_gps*sigma_gps, sigma_gps*sigma_gps, sigma_gps*sigma_gps, sigma_lm*sigma_lm]))
mean_delta_3 = np.zeros(3)
mean_delta_4 = np.zeros(4)
for i in range(num_t_steps):
tt_now = i * deltaT *0.1
u_t = np.array([[-0.128*cos(0.032*i)], [-0.128*sin(0.032*i)], [0.0123]]) ## Control Inputs are s.t. motion is helical
X_init = action_upd(X_init, A_t, B_t, u_t, mean_epsilon, R)
if(np.linalg.norm(L1 - X_init[0:3].squeeze()) < landmark_range):
#In L1's range
z_t = obsv_upd_land(X_init, mean_delta_4, Q_gps_landmark, 1) # z_t is 4*1
mu_init, sigma_init = ext_kalman_update(mu_init, sigma_init, u_t, z_t, A_t, B_t, Q_gps_landmark, R, 1)
elif(np.linalg.norm(L2 - X_init[0:3].squeeze()) < landmark_range):
#In L2's range
z_t = obsv_upd_land(X_init, mean_delta_4, Q_gps_landmark, 2)
mu_init, sigma_init = ext_kalman_update(mu_init, sigma_init, u_t, z_t, A_t, B_t, Q_gps_landmark, R, 2)
elif(np.linalg.norm(L3 - X_init[0:3].squeeze()) < landmark_range):
#In L3's range
z_t = obsv_upd_land(X_init, mean_delta_4, Q_gps_landmark, 3)
mu_init, sigma_init = ext_kalman_update(mu_init, sigma_init, u_t, z_t, A_t, B_t, Q_gps_landmark, R, 3)
elif(np.linalg.norm(L4 - X_init[0:3].squeeze()) < landmark_range):
#In L4's range
z_t = obsv_upd_land(X_init, mean_delta_4, Q_gps_landmark, 4)
mu_init, sigma_init = ext_kalman_update(mu_init, sigma_init, u_t, z_t, A_t, B_t, Q_gps_landmark, R, 4)
else:
#In nobody's range, just do a simple update
z_t = obsv_upd(X_init, C_t, mean_delta_3, Q_gps_only)
mu_init, sigma_init = kalman_update(mu_init, sigma_init, u_t, z_t, A_t, B_t, C_t, Q_gps_only, R)
#print(sigma_init)
#observed_traj[i] = z_t.squeeze()
actual_traj[i] = X_init[0:3].squeeze()
estimated_traj[i] = mu_init[0:3].squeeze()
x_uncert_points, y_uncert_points, z_uncert_points = generate_ellipsoid_points(mu_init, sigma_init)
#x_uncert_points, y_uncert_points = uncer_ellipse_params(mu_init, sigma_init)
uncertainity_ellipse_values[i] = np.column_stack((x_uncert_points, y_uncert_points, z_uncert_points))
fig = go.Figure(data=[go.Scatter3d(
x=actual_traj[:, 0],
y=actual_traj[:, 1],
z=actual_traj[:, 2],
mode='lines',
name='Actual Trajectory'
)])
fig.add_trace(go.Scatter3d(
x=estimated_traj[:, 0],
y=estimated_traj[:, 1],
z=estimated_traj[:, 2],
mode='lines',
line=dict(color='blue', width=2),
name='Extended Kalman Estimation'
))
# Update the layout if needed
fig.update_layout(
title= '3D Line Plots of Actual vs Estimated Trajectory, with 4 landmarks',
scene=dict(aspectmode='data')
)
theta = np.linspace(0, 2*np.pi, 40)
phi = np.linspace(0, np.pi, 20)
theta, phi = np.meshgrid(theta, phi)
r = landmark_range # radius of the sphere
x1s = 150 + r * np.sin(phi) * np.cos(theta)
y1s = r * np.sin(phi) * np.sin(theta)
z1s = 100 + r * np.cos(phi)
x2s = -150 + r * np.sin(phi) * np.cos(theta)
y2s = r * np.sin(phi) * np.sin(theta)
z2s = 100 + r * np.cos(phi)
x3s = r * np.sin(phi) * np.cos(theta)
y3s = -150 + r * np.sin(phi) * np.sin(theta)
z3s = 100 + r * np.cos(phi)
x4s = r * np.sin(phi) * np.cos(theta)
y4s = 150 + r * np.sin(phi) * np.sin(theta)
z4s = 100 + r * np.cos(phi)
# Create scatter3d plot
fig.add_trace(go.Scatter3d(x=x1s.flatten(), y=y1s.flatten(), z=z1s.flatten(), mode='markers', marker=dict(size=2)))
fig.add_trace(go.Scatter3d(x=x2s.flatten(), y=y2s.flatten(), z=z2s.flatten(), mode='markers', marker=dict(size=2)))
fig.add_trace(go.Scatter3d(x=x3s.flatten(), y=y3s.flatten(), z=z3s.flatten(), mode='markers', marker=dict(size=2)))
fig.add_trace(go.Scatter3d(x=x4s.flatten(), y=y4s.flatten(), z=z4s.flatten(), mode='markers', marker=dict(size=2)))
for i in range(0, num_t_steps, 100):
fig.add_trace(go.Scatter3d(
x=uncertainity_ellipse_values[i][:,0],
y=uncertainity_ellipse_values[i][:,1],
z=uncertainity_ellipse_values[i][:,2],
mode='lines',
line=dict(color='gold', width=1),
showlegend=False
))
'''
theta = np.linspace(0, 2 * np.pi, 100)
x_rotated_1 = 150.0 + (landmark_range * np.cos(theta))
x_rotated_2 = -150.0 + (landmark_range * np.cos(theta))
x_rotated_3 = 25.0 + (landmark_range * np.cos(theta))
x_rotated_4 = 0.0 + (landmark_range * np.cos(theta))
y_rotated_1 = 0.0 + (landmark_range * np.sin(theta))
y_rotated_2 = 150.0 + (landmark_range * np.sin(theta))
y_rotated_3 = -150.0 + (landmark_range * np.sin(theta))
fig.add_trace(go.Scatter(x=x_rotated_1,y=y_rotated_1,mode='lines',line=dict(color='black', width=1),showlegend=False))
fig.add_trace(go.Scatter(x=x_rotated_2,y=y_rotated_1,mode='lines',line=dict(color='black', width=1),showlegend=False))
fig.add_trace(go.Scatter(x=x_rotated_4,y=y_rotated_2,mode='lines',line=dict(color='black', width=1),showlegend=False))
fig.add_trace(go.Scatter(x=x_rotated_4,y=y_rotated_3,mode='lines',line=dict(color='black', width=1),showlegend=False))
fig.add_trace(go.Scatter(x=x_rotated_3,y=y_rotated_1,mode='lines',line=dict(color='black', width=1),showlegend=False))
'''
fig.show()
fig.write_html("path_withlandmarks.html")
return 0