es_ekf.py

# Main code for the Coursera SDC Course 2 final project
#
# Author: Trevor Ablett
# University of Toronto Institute for Aerospace Studies

# Assignments Solution Author: Engin Bozkurt

import pickle
import sys
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

from rotations import Quaternion, skew_symmetric
sys.path.append('./data')

#### 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 p3_data.pkl.
################################################################################################
with open('data/p1_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 just 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_li_i.
#
# THIS IS THE CODE YOU WILL MODIFY FOR PART 2 OF THE ASSIGNMENT.
################################################################################################
# This is the correct calibration rotation matrix, corresponding to an euler rotation of 0.05, 0.05, .1.
C_li = np.array([
    [ 0.99376, -0.09722,  0.05466],
    [ 0.09971,  0.99401, -0.04475],
    [-0.04998,  0.04992,  0.9975 ]
])

t_li_i = np.array([0.5, 0.1, 0.5])

lidar.data = (C_li @ lidar.data.T).T + t_li_i


#### 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 = 0.01
var_imu_w = 0.01
var_gnss = 0.1
var_lidar = 35
gravity = 9.81

################################################################################################
# We can also set up some constants that won't change for any iteration of our solver.
################################################################################################
g = np.array([0, 0, -gravity])  # 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.eye(9)  # covariance of estimate

#### 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(sensor_var, p_cov_check, y_k, p_check, v_check, q_check):
    # 3.1 Compute Kalman Gain
    R_cov = sensor_var * np.eye(3)
    K = p_cov_check.dot(h_jac.T.dot(np.linalg.inv(h_jac.dot(p_cov_check.dot(h_jac.T)) + R_cov)))

    # 3.2 Compute error state
    delta_x = K.dot(y_k - p_check)

    # 3.3 Correct predicted state
    p_check = p_check + delta_x[:3]
    v_check = v_check + delta_x[3:6]
    q_check = Quaternion(axis_angle = delta_x[6:]).quat_mult(q_check)

    # 3.4 Compute corrected covariance
    p_cov_check = (np.eye(9) - K.dot(h_jac)).dot(p_cov_check)

    return p_check, v_check, q_check, p_cov_check


#### 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 nominal state with IMU inputs
    Rotation_Mat = Quaternion(*q_est[k - 1]).to_mat()
    p_est[k] = p_est[k - 1] + delta_t * v_est[k - 1] + 0.5 * (delta_t ** 2) * (Rotation_Mat.dot(imu_f.data[k - 1]) + g)
    v_est[k] = v_est[k - 1] + delta_t * (Rotation_Mat.dot(imu_f.data[k - 1]) - g)
    q_est[k] = Quaternion(euler = delta_t * imu_w.data[k - 1]).quat_mult(q_est[k - 1])

    # 1.1 Linearize Motion Model and compute Jacobians
    F = np.eye(9)
    imu = imu_f.data[k - 1].reshape((3, 1))
    F[0:3, 3:6] = delta_t * np.eye(3)
    F[3:6, 6:9] = Rotation_Mat.dot(-skew_symmetric(imu)) * delta_t

    # 2. Propagate uncertainty
    Q = np.eye(6)
    Q[0:3, 0:3] = var_imu_f * Q[0:3, 0:3]
    Q[3:6, 3:6] = var_imu_w * Q[3:6, 3:6]
    Q = (delta_t ** 2) * Q #Integration acceleration to obstain Position
    p_cov[k] = F.dot(p_cov[k - 1]).dot(F.T) + l_jac.dot(Q).dot(l_jac.T)

    # 3. Check availability of GNSS and LIDAR measurements
    for i in range(len(gnss.t)):
        if abs(gnss.t[i] - imu_f.t[k]) < 0.01:
            p_est[k], v_est[k], q_est[k], p_cov[k] = measurement_update(var_gnss, p_cov[k],
                                                    gnss.data[i], p_est[k], v_est[k], q_est[k])
    for i in range(len(lidar.t)):
        if abs(lidar.t[i] - imu_f.t[k]) < 0.01:
            p_est[k], v_est[k], q_est[k], p_cov[k] = measurement_update(var_lidar, p_cov[k],
                                                    lidar.data[i], p_est[k], v_est[k], q_est[k])


#### 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('x [m]')
ax.set_ylabel('y [m]')
ax.set_zlabel('z [m]')
ax.set_title('Estimated Trajectory')
ax.legend()
ax.set_zlim(-1, 5)
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.
################################################################################################
error_fig, ax = plt.subplots(2, 3)
error_fig.suptitle('Error plots')
num_gt = gt.p.shape[0]
p_est_euler = []

# Convert estimated quaternions to euler angles
for q in q_est:
    p_est_euler.append(Quaternion(*q).to_euler())
p_est_euler = np.array(p_est_euler)

# Get uncertainty estimates from P matrix
p_cov_diag_std = np.sqrt(np.diagonal(p_cov, axis1=1, axis2=2))

titles = ['x', 'y', 'z', 'x rot', 'y rot', 'z rot']
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_diag_std[:num_gt, i], 'r--')
    ax[0, i].plot(range(num_gt), -3 * p_cov_diag_std[:num_gt, i], 'r--')
    ax[0, i].set_title(titles[i])

for i in range(3):
    ax[1, i].plot(range(num_gt), gt.r[:, i] - p_est_euler[:num_gt, i])
    ax[1, i].plot(range(num_gt), 3 * p_cov_diag_std[:num_gt, i+6], 'r--')
    ax[1, i].plot(range(num_gt), -3 * p_cov_diag_std[:num_gt, i+6], 'r--')
    ax[1, i].set_title(titles[i+3])
plt.show()