In [1]:
# Import dependencies
from __future__ import division, print_function
%matplotlib inline
import scipy
import time as ti
from filterpy.kalman import unscented_transform, MerweScaledSigmaPoints
from UKF_RST import UnscentedKalmanFilter as UKF
from BicycleTrajectory2D import *
from BicycleUtils import *
from FormatUtils import *
from PlotUtils import *
from DatasetHelper import *
from RealDatasetImporter import *
In [2]:
global_path = '../../bags/experiments/10_09_2017/csv/'
file_name = 'field_half/field_half_preprocessed.csv'
# Import CSV as pandas dataframe and define time as index
data = pd.read_csv(global_path + file_name, index_col=0, parse_dates=True)
data['time_index'] = pd.to_datetime(data['time'])
data = data.set_index('time_index', drop=True, verify_integrity=True)
data['time'] = data.index
di = RealDatasetHelper(data)
# Convert pandas DataFrame into np matrix
linear_a = data[[di.linear_a]].as_matrix()
angular_phi = data[[di.angular_vel_phi]].as_matrix()
angular_delta = data[[di.angular_vel_delta]].as_matrix()
# select time: convert from pandas to numpy and fix initial offset
time = data[[di.time]].as_matrix()
time = time.astype('float64')/1e9
time = time - time[0]
dpi = 150
In [3]:
# Select which velocity to use: wheel encoder of GSP -based:
# [v, v_scale] = [di.real_v, 1.0] # use Wheel encoder velocity, 0.23 scale for old datasets
[v, v_scale] = [di.real_v_gps, 1.0] # use GPS velocity
# Use optical steering angle or IMU steering angle
use_optical_steering = False
sim = di.data.filter(items=[di.real_xf, di.real_xr, di.real_yf, di.real_yr,
di.real_zf, di.real_zr, di.real_za,
di.real_delta if use_optical_steering else di.real_delta_imu,
di.real_psi, di.real_phi]).as_matrix()
sim_view = sim.copy()
# Fix initial offset:
offset_x = sim[0, 0]
offset_y = sim[0, 2]
offset_delta = sim_view[0, 7]
sim_view[:, 0] -= offset_x
sim_view[:, 1] -= offset_x
sim_view[:, 2] -= offset_y
sim_view[:, 3] -= offset_y
sim_view[:, 7] -= offset_delta
# select imputs:
U = di.data.filter(items=[v, di.angular_vel_phi, di.angular_vel_delta]).as_matrix()
U[:, 0] *= v_scale
add_str = ''
if use_optical_steering:
add_str = '_rotatory'
# plot state variables
path_output_simulation = 'experiments/' + file_name.split("/")[0] + add_str + '/'
plot_real_data_state_variables(U=U, sim=sim_view, time=time, file_name=path_output_simulation, dpi=dpi)
In [6]:
sim_imu = di.data.filter(items=[di.real_xf, di.real_xr, di.real_yf, di.real_yr,
di.real_zf, di.real_zr, di.real_za,
di.real_delta_imu,
di.real_psi, di.real_phi]).as_matrix()
sim_rotatory = di.data.filter(items=[di.real_xf, di.real_xr, di.real_yf, di.real_yr,
di.real_zf, di.real_zr, di.real_za,
di.real_delta,
di.real_psi, di.real_phi]).as_matrix()
plot_real_data_delta_comparison(sim_rotatory=sim_rotatory, sim_imu=sim_imu,
time=time, file_name=path_output_simulation, dpi=dpi)
In [4]:
class EKF_sigma_model_fusion(object):
"""Implements an EKF to bicycle model"""
def __init__(self, xs, P, R_std, Q_std, wheel_distance=1.2, dt=0.1, alpha=1.0):
self.w = wheel_distance #Set the distance between the wheels
self.xs = xs #Set the initial state
self.P = P #Set the initial Covariance
self.dt = dt
self.R_std = R_std
self.Q_std = Q_std
self.alpha = alpha
self.K = np.zeros((6, 6)) # Kalman gain
#Set the process noise covariance
self.Q = np.diag([self.Q_std[0], # v
self.Q_std[1], # phi_dot
self.Q_std[2] # delta_dot
])
# Set the measurement noise covariance
self.R = np.diag([self.R_std[0], # xf
self.R_std[1], # xr
self.R_std[2], # yf
self.R_std[3], # yr
self.R_std[4], # zf
self.R_std[5], # zr
self.R_std[6], # za
self.R_std[7], # sigma
self.R_std[8], # psi
self.R_std[9]]) # phi
# Linear relationship H - z = Hx
self.H = np.zeros((10, 6)) # 10 measurements x 6 state variables
[self.H[0, 0], self.H[1, 0]] = [1.0, 1.0] # x
[self.H[2, 1], self.H[3, 1]] = [1.0, 1.0] # y
[self.H[4, 2], self.H[5, 2], self.H[6, 2]] = [1.0, 1.0, 1.0] # z
[self.H[7, 3], self.H[8, 4], self.H[9, 5]] = [1.0, 1.0, 1.0] # sigma - psi - phi
def Fx(self, xs, u):
""" Linearize the system with the Jacobian of the x """
F_result = np.eye(len(xs))
v = u[0]
phi_dot = u[1]
delta_dot = u[2]
sigma = xs[3]
psi = xs[4]
phi = xs[5]
t = self.dt
F04 = -t * v * np.sin(psi)
F14 = t * v * np.cos(psi)
F33 = (2 * t * delta_dot * sigma * self.w) + 1
F43 = (t * v)/np.cos(phi)
F45 = t * sigma * v * np.sin(phi) / np.cos(phi)**2
F_result[0, 4] = F04
F_result[1, 4] = F14
F_result[3, 3] = F33
F_result[4, 3] = F43
F_result[4, 5] = F45
return F_result
def Fu(self, xs, u):
""" Linearize the system with the Jacobian of the u """
v = u[0]
phi_dot = u[1]
delta_dot = u[2]
sigma = xs[3]
psi = xs[4]
phi = xs[5]
t = self.dt
V_result = np.zeros((len(xs), len(u)))
V00 = t * np.cos(psi)
V10 = t * np.sin(psi)
V32 = (t/self.w)*((sigma**2)*(self.w**2) + 1)
V40 = t * sigma / np.cos(phi)
V51 = t
V_result[0, 0] = V00
V_result[1, 0] = V10
V_result[3, 2] = V32
V_result[4, 0] = V40
V_result[5, 1] = V51
return V_result
def f(self, xs, u):
""" Estimate the non-linear state of the system """
v = u[0]
phi_dot = u[1]
delta_dot = u[2]
sigma = xs[3]
psi = xs[4]
phi = xs[5]
t = self.dt
fxu_result = np.zeros((len(xs), 1))
fxu_result[0] = xs[0] + t * v * np.cos(psi)
fxu_result[1] = xs[1] + t * v * np.sin(psi)
fxu_result[2] = xs[2]
fxu_result[3] = xs[3] + (t*phi_dot/self.w)*((sigma**2)*(self.w**2) +1)
fxu_result[4] = xs[4] + t * v * sigma / np.cos(phi)
fxu_result[5] = xs[5] + t * phi_dot
return fxu_result
def h(self, x):
""" takes a state variable and returns the measurement
that would correspond to that state. """
sensor_out = np.zeros((10, 1))
sensor_out[0] = x[0]
sensor_out[1] = x[0]
sensor_out[2] = x[1]
sensor_out[3] = x[1]
sensor_out[4] = x[2]
sensor_out[5] = x[2]
sensor_out[6] = x[2]
sensor_out[7] = x[3] # sigma
sensor_out[8] = x[4] # psi
sensor_out[9] = x[5] # phi
return sensor_out
def Prediction(self, u):
x_ = self.xs
P_ = self.P
self.xs = self.f(x_, u)
self.P = self.alpha * self.Fx(x_, u).dot(P_).dot((self.Fx(x_,u)).T) + \
self.Fu(x_,u).dot(self.Q).dot((self.Fu(x_,u)).T)
def Update(self, z):
"""Update the Kalman Prediction using the meazurement z"""
y = z - self.h(self.xs)
self.K = self.P.dot(self.H.T).dot(np.linalg.inv(self.H.dot(self.P).dot(self.H.T) + self.R))
self.xs = self.xs + self.K.dot(y)
self.P = (np.eye(len(self.xs)) - self.K.dot(self.H)).dot(self.P)
In [5]:
class UKF_Sigma_model_sensor_fusion(object):
def __init__(self, x_init, Q, R, sigma, dt=0.25, w=1.0):
self.fx_filter_vel = 0.0
self.fy_filter_vel = 0.0
self.fz_filter_vel = 0.0
self.fsigma_filter_vel = 0.0
self.fpsi_filter_vel = 0.0
self.fphi_filter_vel = 0.0
self.U_init = []
self.w = w
self.dt = dt
self.t = 0
self.number_state_variables = 6
[self.alpha, self.beta, self.kappa] = [sigma[0], sigma[1], sigma[2]]
self.points = MerweScaledSigmaPoints(n=self.number_state_variables,
alpha=self.alpha, beta=self.beta, kappa=self.kappa)
self.kf = UKF(dim_x=number_state_variables, dim_z=10, dt=self.dt,
fx=self.f_bicycle, hx=self.H_bicycle, points=self.points)
# Q Process Noise Matrix
self.kf.Q = Q
# R Measurement Noise Matrix
self.kf.R = R
self.kf.x = x_init # Initial state
self.kf.P = np.eye(self.number_state_variables) * 10 # Covariance matrix
# Linear relationship H - z = Hx
self.H = np.zeros((10, 6)) # 10 measurements x 6 state variables
[self.H[0, 0], self.H[1, 0]] = [1.0, 1.0] # x
[self.H[2, 1], self.H[3, 1]] = [1.0, 1.0] # y
[self.H[4, 2], self.H[5, 2], self.H[6, 2]] = [1.0, 1.0, 1.0] # z
[self.H[7, 3], self.H[8, 4], self.H[9, 5]] = [1.0, 1.0, 1.0] # sigma - psi - phi
def fx_filter(self, x, t):
return self.fx_filter_vel
def fy_filter(self, y, t):
return self.fy_filter_vel
def fz_filter(self, y, t):
return self.fz_filter_vel
def fsigma_filter(self, y, t):
return self.fsigma_filter_vel
def fpsi_filter(self, yaw, t):
return self.fpsi_filter_vel
def fphi_filter(self, yaw, t):
return self.fphi_filter_vel
def rk4(self, y, x, dx, f):
k1 = dx * f(y, x)
k2 = dx * f(y + 0.5*k1, x + 0.5*dx)
k3 = dx * f(y + 0.5*k2, x + 0.5*dx)
k4 = dx * f(y + k3, x + dx)
return y + (k1 + 2*k2 + 2*k3 + k4) / 6.
def f_bicycle(self, x, dt, U=None):
if U is None:
U = self.U_init
x_out = x.copy()
[x_ini, y_ini, z_ini, sigma_ini, psi_ini, phi_ini] = x.copy()
v_ini = U[0]
phi_dot = U[1]
delta_dot = U[2]
#Solve diff equation by approximation
x = self.rk4(x_ini, self.t, self.dt, self.fx_filter)
y = self.rk4(y_ini, self.t, self.dt, self.fy_filter)
z = self.rk4(z_ini, self.t, self.dt, self.fz_filter)
sigma = self.rk4(sigma_ini, self.t, self.dt, self.fsigma_filter)
psi = self.rk4(psi_ini, self.t, self.dt, self.fpsi_filter)
phi = self.rk4(phi_ini, self.t, self.dt, self.fphi_filter)
self.fx_filter_vel = math.cos(psi) * v_ini
self.fy_filter_vel = math.sin(psi) * v_ini
self.fz_filter_vel = 0
self.fsigma_filter_vel = (phi_dot / self.w)*(1 + (self.w**2)*(sigma_ini**2))
self.fpsi_filter_vel = (v_ini * sigma_ini) / math.cos(phi_ini)
self.fphi_filter_vel = phi_dot
x_out[0] = x
x_out[1] = y
x_out[2] = z
x_out[3] = sigma
x_out[4] = psi
x_out[5] = phi
return x_out
def H_bicycle(self, x):
""" takes a state variable and returns the measurement
that would correspond to that state. """
sensor_out = np.zeros(10)
sensor_out = self.H.dot(x)
return sensor_out
def disable_GPS(self):
[self.H[0, 0], self.H[1, 0]] = [0.0, 0.0] # x
[self.H[2, 1], self.H[3, 1]] = [0.0, 0.0] # y
def enable_GPS(self):
[self.H[0, 0], self.H[1, 0]] = [1.0, 1.0] # x
[self.H[2, 1], self.H[3, 1]] = [1.0, 1.0] # y
In [6]:
[t, wheel_distance, number_state_variables] = [0.0, 1.1, 6]
z = sim.copy()
# set UTM offset at first measurement
utm_offset_x = z[0, 0]
utm_offset_y = z[0, 2]
utm_offset_z = z[0, 4]
X_init = np.array([utm_offset_x, utm_offset_y, utm_offset_z, 0.0, 0.0, 0.0]) # [x, y, z, sigma, psi, phi]
alpha = 1.06
# covariance matrix
P = np.eye(number_state_variables) * 10
dt = 1.0/5.0 # Slower Sample Rate of the Measurements is 5Hz
# defining how bad things may goes, take max acceleratin value x margin
margin = 2
# EKF parameters ---------------------------------------------------------------------------------------
# process noise covariance Q Maximum change (acceleration) for given dataset
max_acc_v = float(di.data[[di.real_v]].diff().max()) * margin
max_acc_phi_dot = float(di.data[[di.angular_vel_phi]].diff().max()) * margin
max_acc_delta_dot = float(di.data[[di.angular_vel_delta]].diff().max()) * margin
sigma_v = (max_acc_v*dt)**2
sigma_phi_dot = (max_acc_phi_dot*dt)**2
sigma_delta_dot = (max_acc_delta_dot*dt)**2
Q_std = [sigma_v, sigma_phi_dot, sigma_delta_dot] # v, phi_dot, delta_dot
# measurement noise covariance R
R_std = [0.1**2, 0.1**2, # x
0.1**2, 0.1**2, # y
0.1**2, 0.1**2, 0.1**2, # z
0.001**2, 0.001**2, 0.005**2] # delta - psi - phi
[offset_psi, offset_phi, offset_delta] = [0.0, 0.0, 0.0]
filter_ekf = EKF_sigma_model_fusion(X_init, P, R_std=R_std, Q_std=Q_std, wheel_distance=wheel_distance, dt=dt, alpha=alpha)
#-------------------------------------------------------------------------------------------------------
# UKF parameters ---------------------------------------------------------------------------------------
# Q Process Noise Matrix
max_acc_x = float(di.data[[di.real_xf]].diff().max()) * margin
max_acc_y = float(di.data[[di.real_yf]].diff().max()) * margin
max_acc_z = float(di.data[[di.real_zf]].diff().max()) * margin
max_acc_psi = float(di.data[[di.real_psi]].diff().max()) * margin
max_acc_phi = float(di.data[[di.real_phi]].diff().max()) * margin
max_acc_delta = float(di.data[[di.real_delta]].diff().max()) * margin
Q_ukf = np.diag([max_acc_x**2, max_acc_y**2, max_acc_z**2, max_acc_delta**2, max_acc_psi**2, max_acc_delta**2])
# measurement noise covariance R [xf, xr, yf, yr, zf, zr, za, delta, psi, phi]
# R_std = [8.5**2, 8.5**2, 8.5**2, 1.8**2, 8.5**2, 1.8**2] # [x, y, z, sigma, psi, phi]
R_ukf = np.diag([1.5**2, 1.5**2, # x
1.5**2, 1.5**2, # y
1.5**2, 1.5**2, 1.5**2, # z
0.05**2, 0.05**2, 0.05**2]) # delta - psi - phi
''' Sigma point'''
sigma = [alpha, beta, kappa] = [0.8, 2.0, -2.0]
filter_ukf = UKF_Sigma_model_sensor_fusion(x_init=X_init, Q=Q_ukf, R=R_ukf, sigma=sigma, dt=dt, w=wheel_distance)
#-------------------------------------------------------------------------------------------------------
Ut = np.array([0.0, 0.0, 0.0]) # [v, phi_dot, delta_dot]
xs_ekf = np.zeros((len(time), number_state_variables))
xs_ukf = np.zeros((len(time), number_state_variables))
z_t = np.zeros((10, 1))
t = range(1, len(time))
diff_t_ekf = np.zeros((len(time), 1))
diff_t_ukf = np.zeros((len(time), 1))
dt_real = dt
for i in range(0, len(time)):
if i > 1:
dt_real = float(time[i] - time[i-1]) # time: nano to seg
filter_ekf.dt = dt_real
filter_ukf.kf._dt = dt_real
# update U
Ut[0] = float(U[i, 0])
Ut[1] = float(U[i, 1])
Ut[2] = float(U[i, 2])
#Update measurements [xf, xr, yf, yr, zf, zr, za, delta, psi, phi]
z_t[0] = z[i, 0] # xf
z_t[1] = z[i, 1] # xr
z_t[2] = z[i, 2] # yf
z_t[3] = z[i, 3] # yr
z_t[4] = z[i, 4] # zf
z_t[5] = z[i, 5] # zr
z_t[6] = z[i, 6] # za
z_t[7] = np.tan(z[i, 7])/wheel_distance # sigma
z_t[8] = z[i, 8] # psi
z_t[9] = z[i, 9] # phi
# EKF -----------------------
xs_ekf[i] = filter_ekf.xs.T
time_start = ti.time()
filter_ekf.Prediction(Ut)
filter_ekf.Update(z_t)
time_end = ti.time()
diff_t_ekf[i] = time_end - time_start
# UKF -----------------------
xs_ukf[i,:] = filter_ukf.kf.x
time_start = ti.time()
filter_ukf.kf.predict(dt=dt_real, fx_args=(U[i]))
filter_ukf.kf.update(z[i])
time_end = ti.time()
diff_t_ukf[i] = time_end - time_start
# update delta based on sigma
xs_ekf[:, 3] = np.arctan2(xs_ekf[:, 3], 1/wheel_distance) # delta
xs_ukf[:, 3] = np.arctan2(xs_ukf[:, 3], 1/wheel_distance) # delta
# Fix initial offset (for visualization):
xs_view_ekf = xs_ekf.copy()
xs_view_ekf[:, 0] -= offset_x
xs_view_ekf[:, 1] -= offset_y
xs_view_ukf = xs_ukf.copy()
xs_view_ukf[:, 0] -= offset_x
xs_view_ukf[:, 1] -= offset_y
z_view = z.copy()
z_view[:, 0] -= offset_x
z_view[:, 1] -= offset_x
z_view[:, 2] -= offset_y
z_view[:, 3] -= offset_y
In [7]:
init_pos = 0
samples = 1350
final_pos = init_pos + samples
xs_view_ekf_filter = xs_view_ekf[range(init_pos, final_pos), :]
xs_view_ukf_filter = xs_view_ukf[range(init_pos, final_pos), :]
z_view_filter = z_view[range(init_pos, final_pos), :]
time_filter = time[range(init_pos, final_pos),]
path_output_filter = 'filters/IMU_vs_rotatory_sensor/' + file_name.split("/")[0] + add_str + "/"
plot_comparison_real_data(xs_ekf=xs_view_ekf_filter, xs_ukf=xs_view_ukf_filter,
sim=z_view_filter, time=time_filter,
file_name=path_output_filter, dpi=dpi, format='png')
In [ ]: