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es_ekf.py
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# Starter code for the Coursera SDC Course 2 final project.
#
# Author: Trevor Ablett and Jonathan Kelly
# University of Toronto Institute for Aerospace Studies
import pickle
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from rotations import angle_normalize, rpy_jacobian_axis_angle, skew_symmetric, Quaternion
#### 1. Data ###################################################################################
################################################################################################
# This is where you will load the data from the pickle files. For parts 1 and 2, you will use
# p1_data.pkl. For Part 3, you will use pt3_data.pkl.
################################################################################################
with open('data/pt1_data.pkl', 'rb') as file:
data = pickle.load(file)
################################################################################################
# Each element of the data dictionary is stored as an item from the data dictionary, which we
# will store in local variables, described by the following:
# gt: Data object containing ground truth. with the following fields:
# a: Acceleration of the vehicle, in the inertial frame
# v: Velocity of the vehicle, in the inertial frame
# p: Position of the vehicle, in the inertial frame
# alpha: Rotational acceleration of the vehicle, in the inertial frame
# w: Rotational velocity of the vehicle, in the inertial frame
# r: Rotational position of the vehicle, in Euler (XYZ) angles in the inertial frame
# _t: Timestamp in ms.
# imu_f: StampedData object with the imu specific force data (given in vehicle frame).
# data: The actual data
# t: Timestamps in ms.
# imu_w: StampedData object with the imu rotational velocity (given in the vehicle frame).
# data: The actual data
# t: Timestamps in ms.
# gnss: StampedData object with the GNSS data.
# data: The actual data
# t: Timestamps in ms.
# lidar: StampedData object with the LIDAR data (positions only).
# data: The actual data
# t: Timestamps in ms.
################################################################################################
gt = data['gt']
imu_f = data['imu_f']
imu_w = data['imu_w']
gnss = data['gnss']
lidar = data['lidar']
################################################################################################
# Let's plot the ground truth trajectory to see what it looks like. When you're testing your
# code later, feel free to comment this out.
################################################################################################
gt_fig = plt.figure()
ax = gt_fig.add_subplot(111, projection='3d')
ax.plot(gt.p[:,0], gt.p[:,1], gt.p[:,2])
ax.set_xlabel('x [m]')
ax.set_ylabel('y [m]')
ax.set_zlabel('z [m]')
ax.set_title('Ground Truth trajectory')
ax.set_zlim(-1, 5)
plt.show()
################################################################################################
# Remember that our LIDAR data is actually just a set of positions estimated from a separate
# scan-matching system, so we can insert it into our solver as another position measurement,
# just as we do for GNSS. However, the LIDAR frame is not the same as the frame shared by the
# IMU and the GNSS. To remedy this, we transform the LIDAR data to the IMU frame using our
# known extrinsic calibration rotation matrix C_li and translation vector t_i_li.
#
# THIS IS THE CODE YOU WILL MODIFY FOR PART 2 OF THE ASSIGNMENT.
################################################################################################
# Correct calibration rotation matrix, corresponding to Euler RPY angles (0.05, 0.05, 0.1).
C_li = np.array([
[ 0.99376, -0.09722, 0.05466],
[ 0.09971, 0.99401, -0.04475],
[-0.04998, 0.04992, 0.9975 ]
])
# Incorrect calibration rotation matrix, corresponding to Euler RPY angles (0.05, 0.05, 0.05).
# C_li = np.array([
# [ 0.9975 , -0.04742, 0.05235],
# [ 0.04992, 0.99763, -0.04742],
# [-0.04998, 0.04992, 0.9975 ]
# ])
t_i_li = np.array([0.5, 0.1, 0.5])
# Transform from the LIDAR frame to the vehicle (IMU) frame.
lidar.data = (C_li @ lidar.data.T).T + t_i_li
#### 2. Constants ##############################################################################
################################################################################################
# Now that our data is set up, we can start getting things ready for our solver. One of the
# most important aspects of a filter is setting the estimated sensor variances correctly.
# We set the values here.
################################################################################################
var_imu_f = 0.10
var_imu_w = 0.25
var_gnss = 0.01
var_lidar = 1.00
################################################################################################
# We can also set up some constants that won't change for any iteration of our solver.
################################################################################################
g = np.array([0, 0, -9.81]) # gravity
l_jac = np.zeros([9, 6])
l_jac[3:, :] = np.eye(6) # motion model noise jacobian
h_jac = np.zeros([3, 9])
h_jac[:, :3] = np.eye(3) # measurement model jacobian
#### 3. Initial Values #########################################################################
################################################################################################
# Let's set up some initial values for our ES-EKF solver.
################################################################################################
p_est = np.zeros([imu_f.data.shape[0], 3]) # position estimates
v_est = np.zeros([imu_f.data.shape[0], 3]) # velocity estimates
q_est = np.zeros([imu_f.data.shape[0], 4]) # orientation estimates as quaternions
p_cov = np.zeros([imu_f.data.shape[0], 9, 9]) # covariance matrices at each timestep
# Set initial values.
p_est[0] = gt.p[0]
v_est[0] = gt.v[0]
q_est[0] = Quaternion(euler=gt.r[0]).to_numpy()
p_cov[0] = np.zeros(9) # covariance of estimate
gnss_i = 0
lidar_i = 0
#### 4. Measurement Update #####################################################################
################################################################################################
# Since we'll need a measurement update for both the GNSS and the LIDAR data, let's make
# a function for it.
################################################################################################
def measurement_update(h_jac,sensor_var, p_cov_check, y_k, p_check, v_check, q_check):
# 3.1 Compute Kalman Gain
i=np.linalg.inv(h_jac@p_cov_check@h_jac.T+sensor_var)
kal=p_cov_check@h_jac.T@i
# 3.2 Compute error state
delx=kal@(y_k-p_check)
delp=delx[:3]
delv=delx[3:6]
delq=delx[6:9]
# 3.3 Correct predicted state
p_hat=p_check+delp
v_hat=v_check+delv
q_hat=Quaternion(axis_angle=delq).quat_mult(q_check, out='np')
# 3.4 Compute corrected covariance
I=np.identity(9)
p_cov_hat=(I-kal@h_jac)@p_cov_check
return p_hat, v_hat, q_hat, p_cov_hat
#### 5. Main Filter Loop #######################################################################
################################################################################################
# Now that everything is set up, we can start taking in the sensor data and creating estimates
# for our state in a loop.
################################################################################################
for k in range(1, imu_f.data.shape[0]): # start at 1 b/c we have initial prediction from gt
delta_t = imu_f.t[k] - imu_f.t[k - 1]
# 1. Update state with IMU inputs
vf=var_imu_f
vw=var_imu_w
Q=(delta_t**2)*np.diag([vf,vf,vf,vw,vw,vw])
# 1.1 Linearize the motion model and compute Jacobians
f=imu_f.data[k-1]
w=imu_w.data[k-1]
C_ns=Quaternion(C_ns*q_est[k-1]).to_mat()
p_check=p_est[k-1]+delta_t*v_est[k-1]+(delta_t**2/2)*(C_ns*f+g)
v_check=v_est[k-1]+delta_t*(C_ns*f+g)
q_check=Quaternion(axis_angle=w*delta_t).quat_mult(q_est[k-1])
# 2. Propagate uncertainty
F=np.identity(9)
F[0:3,3:6]=delta_t*np.identity(3)
F[3:6,6:9]=((-C_ns@f).reshape(3,1))*delta_t
L=np.zeros((9,6))
L[3:6,0:3]=np.eye(3)
L[6:9,3:6]=np.eye(3)
p_cov_check=F@[email protected]+L@[email protected]
# 3. Check availability of GNSS and LIDAR measurements
if gnss_i<49:
if imu_f.t[k-1]==gnss.t[gnss_i]:
gavail=True
else:
gavail=False
if imu_f.t[k-1]==lidar.t[lidar_i]:
liavail=True
else:
liavail=False
p_est[k]=p_check
v_est[k]=v_check
q_est[k]=q_check
p_cov[k]=p_cov_check
if gavail and liavail:
y_k=gnss.data[gnss_i]
p_est[k],v_est[k],q_est[k],p_cov[k]=measurement_update(h_jac,gnss_var, p_cov_check, y_k, p_check, v_check, q_check)
gnss_i+=1
y_k=lidar.data[lidar_i]
p_est[k],v_est[k],q_est[k],p_cov[k]=measurement_update(h_jac,lidar_var, p_cov_check, y_k, p_check, v_check, q_check)
lidar_i+=1
if not gavail and liavail:
y_k=lidar.data[lidar_i]
p_est[k],v_est[k],q_est[k],p_cov[k]=measurement_update(h_jac,lidar_var, p_cov_check, y_k, p_check, v_check, q_check)
lidar_i+=1
if not liavail and gavail:
y_k=gnss.data[gnss_i]
p_est[k],v_est[k],q_est[k],p_cov[k]=measurement_update(h_jac,gnss_var, p_cov_check, y_k, p_check, v_check, q_check)
gnss_i+=1
# Update states (save)
#### 6. Results and Analysis ###################################################################
################################################################################################
# Now that we have state estimates for all of our sensor data, let's plot the results. This plot
# will show the ground truth and the estimated trajectories on the same plot. Notice that the
# estimated trajectory continues past the ground truth. This is because we will be evaluating
# your estimated poses from the part of the trajectory where you don't have ground truth!
################################################################################################
est_traj_fig = plt.figure()
ax = est_traj_fig.add_subplot(111, projection='3d')
ax.plot(p_est[:,0], p_est[:,1], p_est[:,2], label='Estimated')
ax.plot(gt.p[:,0], gt.p[:,1], gt.p[:,2], label='Ground Truth')
ax.set_xlabel('Easting [m]')
ax.set_ylabel('Northing [m]')
ax.set_zlabel('Up [m]')
ax.set_title('Ground Truth and Estimated Trajectory')
ax.set_xlim(0, 200)
ax.set_ylim(0, 200)
ax.set_zlim(-2, 2)
ax.set_xticks([0, 50, 100, 150, 200])
ax.set_yticks([0, 50, 100, 150, 200])
ax.set_zticks([-2, -1, 0, 1, 2])
ax.legend(loc=(0.62,0.77))
ax.view_init(elev=45, azim=-50)
plt.show()
################################################################################################
# We can also plot the error for each of the 6 DOF, with estimates for our uncertainty
# included. The error estimates are in blue, and the uncertainty bounds are red and dashed.
# The uncertainty bounds are +/- 3 standard deviations based on our uncertainty (covariance).
################################################################################################
error_fig, ax = plt.subplots(2, 3)
error_fig.suptitle('Error Plots')
num_gt = gt.p.shape[0]
p_est_euler = []
p_cov_euler_std = []
# Convert estimated quaternions to euler angles
for i in range(len(q_est)):
qc = Quaternion(*q_est[i, :])
p_est_euler.append(qc.to_euler())
# First-order approximation of RPY covariance
J = rpy_jacobian_axis_angle(qc.to_axis_angle())
p_cov_euler_std.append(np.sqrt(np.diagonal(J @ p_cov[i, 6:, 6:] @ J.T)))
p_est_euler = np.array(p_est_euler)
p_cov_euler_std = np.array(p_cov_euler_std)
# Get uncertainty estimates from P matrix
p_cov_std = np.sqrt(np.diagonal(p_cov[:, :6, :6], axis1=1, axis2=2))
titles = ['Easting', 'Northing', 'Up', 'Roll', 'Pitch', 'Yaw']
for i in range(3):
ax[0, i].plot(range(num_gt), gt.p[:, i] - p_est[:num_gt, i])
ax[0, i].plot(range(num_gt), 3 * p_cov_std[:num_gt, i], 'r--')
ax[0, i].plot(range(num_gt), -3 * p_cov_std[:num_gt, i], 'r--')
ax[0, i].set_title(titles[i])
ax[0,0].set_ylabel('Meters')
for i in range(3):
ax[1, i].plot(range(num_gt), \
angle_normalize(gt.r[:, i] - p_est_euler[:num_gt, i]))
ax[1, i].plot(range(num_gt), 3 * p_cov_euler_std[:num_gt, i], 'r--')
ax[1, i].plot(range(num_gt), -3 * p_cov_euler_std[:num_gt, i], 'r--')
ax[1, i].set_title(titles[i+3])
ax[1,0].set_ylabel('Radians')
plt.show()
#### 7. Submission #############################################################################
################################################################################################
# Now we can prepare your results for submission to the Coursera platform. Uncomment the
# corresponding lines to prepare a file that will save your position estimates in a format
# that corresponds to what we're expecting on Coursera.
################################################################################################
# Pt. 1 submission
p1_indices = [9000, 9400, 9800, 10200, 10600]
p1_str = ''
for val in p1_indices:
for i in range(3):
p1_str += '%.3f ' % (p_est[val, i])
with open('pt1_submission.txt', 'w') as file:
file.write(p1_str)
# Pt. 2 submission
# p2_indices = [9000, 9400, 9800, 10200, 10600]
# p2_str = ''
# for val in p2_indices:
# for i in range(3):
# p2_str += '%.3f ' % (p_est[val, i])
# with open('pt2_submission.txt', 'w') as file:
# file.write(p2_str)
# Pt. 3 submission
# p3_indices = [6800, 7600, 8400, 9200, 10000]
# p3_str = ''
# for val in p3_indices:
# for i in range(3):
# p3_str += '%.3f ' % (p_est[val, i])
# with open('pt3_submission.txt', 'w') as file:
# file.write(p3_str)