In [3]:
# Import dependencies
from __future__ import division, print_function
%matplotlib inline
import scipy
import sympy
from sympy import Symbol, symbols, Matrix, sin, cos, latex
from sympy.interactive import printing
printing.init_printing()
sympy.init_printing(use_latex="mathjax", fontsize='16pt')
from BicycleTrajectory2D import *
from BicycleUtils import *
from FormatUtils import *
from PlotUtils import *
In [4]:
[N, dt, wheel_distance] = [300, 0.05, 1.1] # simulation parameters
add_noise = True # Enable/disable gaussian noise
# Define initial state --------------------------------------------------------
delta = math.radians(6) # steering angle
phi = math.radians(0) # Lean angle
X_init = np.array([1.0, 3.0, 0.0, np.tan(delta)/wheel_distance, 0.0, phi]) # [x, y, z, sigma, psi, phi]
# Define constant inputs ------------------------------------------------------
U_init = np.array([1.0, 0.01, 0.01]) # [v, phi_dot, delta_dot]
# Define standard deviation for gaussian noise model --------------------------
# [xf, xr, yf, yr, zf, zr, za, delta, psi, phi]
if add_noise:
noise = [0.5, 0.5, 0.5, 0.5, 0.1, 0.1, 0.1, 0.01, 0.01, 0.01]
else:
noise = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
# Create object simulator ------------------------------------------------------
bike = BicycleTrajectory2D(X_init=X_init, U_init=U_init, noise=noise)
# Simulate path ----------------------------------------------------------------
(gt_sim, zs_sim, time) = bike.simulate_path(N=N, dt=dt)
# Plot simulation results
#plot_results(xs=[], zs_sim=zs_sim, gt_sim=gt_sim, time=time, plot_xs=False)
In [5]:
x, y, v, psi, phi, delta, time, w, sigma, z = symbols('x y v \psi \phi \delta T w sigma z')
delta_dot, v_dot, phi_dot = symbols('delta_dot v_dot phi_dot')
fxu = Matrix([[x + time*v*sympy.cos(psi)],
[y + time*v*sympy.sin(psi)],
[z],
[sigma + (time*delta_dot/w)*(1+(w**2)*(sigma**2))],
[psi + time*v*sigma/sympy.cos(phi)],
[phi + time*phi_dot]
])
state = Matrix([x, y, z, sigma, psi, phi]) # Define state
F = fxu.jacobian(state) # Compute Jacobian of F with respecto to states
inputs = Matrix([v, phi_dot, delta_dot]) # Define inputs
V = fxu.jacobian(inputs) # Compute Jacobian of F with respecto to inputs
In [6]:
fxu
#print (latex(fxu))
Out[6]:
In [7]:
F
#print (latex(F))
Out[7]:
In [8]:
V
#print (latex(V))
Out[8]:
In [9]:
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 *0.0 #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
self.K = np.eye(6)*0.5 # 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)
Execute EKF
In [10]:
np.random.seed(850)
file_name = "filters/EKF/math_model/"
[N, dt, wheel_distance, number_state_variables] = [300, 0.05, 1.1, 6]
delta = math.radians(6)
phi = math.radians(0)
#%prun some_useless_slow_function()
U_init = np.array([1.0, 0.01, 0.01]) # [v, phi_dot, delta_dot]
X_init = np.array([1.0, 3.0, 0.0, np.tan(delta)/wheel_distance, 0.0, phi]) # [x, y, z, sigma, psi, phi]
# noise = [xf, xr, yf, yr, zf, zr, za, delta, psi, phi]
if add_noise:
noise = [0.5, 0.5, 0.5, 0.5, 0.1, 0.1, 0.1, 0.01, 0.01, 0.01]
file_name += "noise/"
else:
noise = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
file_name += "no_noise/"
bike = BicycleTrajectory2D(X_init=X_init, U_init=U_init, w=wheel_distance, noise=noise)
(gt_sim, zs_sim, time_t) = bike.simulate_path(N=N, dt=dt)
alpha = 1.0
# covariance matrix
P = np.eye(number_state_variables) * 1e0
# Input noise covariance M
M_std = [(0.001)**2, (0.001)**2, (0.001)**2 ] # v, phi_dot, delta_dot
# Measurement noise covariance R
# [xf, xr, yf, yr, zf, zr, za, delta, psi, phi]
R_std = [0.8**2, 0.8**2, # x
0.8**2, 0.8**2, # y
0.5**2, 0.5**2, 0.5**2, # z
0.5**2, 0.4**2, 1.8**2] # delta - psi - phi
filter_ekf = EKF_sigma_model_fusion(X_init, P, R_std=R_std, Q_std=M_std, wheel_distance=wheel_distance, dt=dt, alpha=alpha)
xs = np.zeros((N, number_state_variables))
ps = np.zeros((N, number_state_variables, number_state_variables))
PU = np.zeros((N, number_state_variables))
KU = np.zeros((N, number_state_variables))
time_t = np.zeros((N, 1))
t = 0
z_t = np.zeros((10, 1))
for i in range(N):
P = filter_ekf.P
K = filter_ekf.K
PU[i] = [P[0,0], P[1,1], P[2,2], P[3,3], P[4,4], P[5,5]]
KU[i] = [K[0,0], K[1,1], K[2,2], K[3,3], K[4,4], K[5,5]]
xs[i] = filter_ekf.xs.T
xs[i, 3] = np.arctan2(xs[i, 3], 1/wheel_distance) # sigma to delta conversion
# predict
filter_ekf.Prediction(U_init)
# update measurements [xf, xr, yf, yr, zf, zr, za, delta, psi, phi]
z_t[0] = zs_sim[i].xf
z_t[1] = zs_sim[i].xr
z_t[2] = zs_sim[i].yf
z_t[3] = zs_sim[i].yr
z_t[4] = zs_sim[i].zf
z_t[5] = zs_sim[i].zr
z_t[6] = zs_sim[i].za
z_t[7] = np.tan(zs_sim[i].delta)/wheel_distance # sigma
z_t[8] = zs_sim[i].psi # psi
z_t[9] = zs_sim[i].phi # phi
filter_ekf.Update(z_t)
cov = np.array([[P[0, 0], P[2, 0]],
[P[0, 2], P[2, 2]]])
mean = (xs[i, 0], xs[i, 1])
#plot_covariance_ellipse(mean, cov, fc='g', std=3, alpha=0.3, title="covariance")
time_t[i] = t
t += dt
filter_ekf.time_t = t
In [11]:
filter_name = 'EKF'
(gt, zs) = convert_object_to_array(gt_sim, zs_sim)
plot_filter_results(xs, gt, zs, time_t, file_name, filter_name)
Plot Kalman gain and process covariance
In [12]:
plot_EKF_gain_covariance(time_t, KU, PU, file_name, autoscale_axis=True)
In [ ]: