Extended-Kalman-Filter-CTRA.py

#!/usr/bin/env python
# coding: utf-8

# In[1]:


import numpy as np
get_ipython().run_line_magic('matplotlib', 'inline')
import matplotlib.dates as mdates
import matplotlib.pyplot as plt
import time
from scipy.stats import norm
from sympy import Symbol, symbols, Matrix, sin, cos
from sympy import init_printing
from sympy.utilities.codegen import codegen
init_printing(use_latex=True)


# # Extended Kalman Filter Implementation for Constant Turn Rate and Acceleration (CTRA) Vehicle Model in Python

# ![Extended Kalman Filter Step](Extended-Kalman-Filter-Step.png)

# [Wikipedia](http://en.wikipedia.org/wiki/Extended_Kalman_filter) writes: In the extended Kalman filter, the state transition and observation models need not be linear functions of the state but may instead be differentiable functions.
# 
# $\boldsymbol{x}_{k} = g(\boldsymbol{x}_{k-1}, \boldsymbol{u}_{k-1}) + \boldsymbol{w}_{k-1}$
# 
# $\boldsymbol{z}_{k} = h(\boldsymbol{x}_{k}) + \boldsymbol{v}_{k}$
# 
# Where $w_k$ and $v_k$ are the process and observation noises which are both assumed to be zero mean Multivariate Gaussian noises with covariance matrix $Q$ and $R$ respectively.
# 
# The function $g$ can be used to compute the predicted state from the previous estimate and similarly the function $h$ can be used to compute the predicted measurement from the predicted state. However, $g$ and $h$ cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the Jacobian matrix) is computed.
# 
# At each time step, the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes the non-linear function around the current estimate.

# Situation covered: You have a velocity sensor, which measures the vehicle speed ($v$) in heading direction ($\psi$) and a yaw rate sensor ($\dot \psi$) which both have to fused with the position ($x$ & $y$) from a GPS sensor.

# ## State Vector - Constant Turn Rate and Acceleration Vehicle Model (CTRA)

# Constant Turn Rate, Constant Velocity Model for a vehicle ![CTRV Model](CTRV-Model.png)
# 
# $$x_k= \left[ \matrix{ x \\ y \\ \psi \\ v \\ \dot\psi \\ a} \right] = \left[ \matrix{ \text{Position X} \\ \text{Position Y} \\ \text{Heading} \\ \text{Velocity} \\ \text{Yaw Rate} \\ \text{longitudinal acceleration}} \right]$$

# In[2]:


numstates=6 # States


# In[3]:


dt = 1.0/50.0 # Sample Rate of the Measurements is 50Hz
dtGPS=1.0/10.0 # Sample Rate of GPS is 10Hz


# All symbolic calculations are made with [Sympy](http://nbviewer.ipython.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-5-Sympy.ipynb). Thanks!

# In[4]:


vs, psis, dpsis, dts, xs, ys, lats, lons, axs = symbols('v \psi \dot\psi T x y lat lon a')

gs = Matrix([[xs + (1 / dpsis**2) * ((vs*dpsis + axs * dpsis * dts) * sin(psis + dpsis* dts) + axs * cos(psis + dpsis * dts) - vs * dpsis * sin(psis) - axs * cos(psis))],    
             [ys + (1 / dpsis**2) * ((-vs*dpsis - axs * dpsis * dts) * cos(psis + dpsis* dts) + axs * sin(psis + dpsis * dts) + vs * dpsis * cos(psis) - axs * sin(psis))],
             [psis+dpsis*dts],
             [axs*dts + vs],
             [dpsis],
             [axs]])
state = Matrix([xs,ys,psis,vs,dpsis,axs])


# ## Dynamic Matrix

# This formulas calculate how the state is evolving from one to the next time step

# In[5]:


gs


# ### Calculate the Jacobian of the Dynamic Matrix with respect to the state vector

# In[6]:


state


# In[7]:


gs.jacobian(state)


# In[8]:


gs.jacobian(state)


# It has to be computed on every filter step because it consists of state variables!
# 
# To Sympy Team: A `.to_python` and `.to_c` and `.to_matlab` whould be nice to generate code, like it already works with `print latex()`.

# ## Initial Uncertainty $P_0$

# Initialized with $0$ means you are pretty sure where the vehicle starts

# In[9]:


P = np.diag([1000.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0])
print(P, P.shape)


# ## Process Noise Covariance Matrix Q

# "*The state uncertainty model models the disturbances which excite the linear system. Conceptually, it estimates how bad things can get when the system is run open loop for a given period of time.*" - Kelly, A. (1994). A 3D state space formulation of a navigation Kalman filter for autonomous vehicles, (May). Retrieved from http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA282853

# In[10]:


sGPS     = 0.5*8.8*dt**2  # assume 8.8m/s2 as maximum acceleration, forcing the vehicle
sCourse  = 0.1*dt # assume 0.1rad/s as maximum turn rate for the vehicle
sVelocity= 8.8*dt # assume 8.8m/s2 as maximum acceleration, forcing the vehicle
sYaw     = 1.0*dt # assume 1.0rad/s2 as the maximum turn rate acceleration for the vehicle
sAccel   = 0.5

Q = np.diag([sGPS**2, sGPS**2, sCourse**2, sVelocity**2, sYaw**2, sAccel**2])
print(Q, Q.shape)


# In[11]:


fig = plt.figure(figsize=(5, 5))
im = plt.imshow(Q, interpolation="none", cmap=plt.get_cmap('binary'))
plt.title('Process Noise Covariance Matrix $Q$')
ylocs, ylabels = plt.yticks()
# set the locations and labels of the yticks
plt.yticks(np.arange(6),('$x$', '$y$', '$\psi$', '$v$', '$\dot \psi$', '$a$'), fontsize=22)

xlocs, xlabels = plt.xticks()

# set the locations and labels of the yticks
plt.xticks(np.arange(6),('$x$', '$y$', '$\psi$', '$v$', '$\dot \psi$', '$a$'), fontsize=22)

plt.xlim([-0.5,5.5])
plt.ylim([5.5, -0.5])

from mpl_toolkits.axes_grid1 import make_axes_locatable
divider = make_axes_locatable(plt.gca())
cax = divider.append_axes("right", "5%", pad="3%")
plt.colorbar(im, cax=cax);


# ## Real Measurements

# In[12]:


#path = './../RaspberryPi-CarPC/TinkerDataLogger/DataLogs/2014/'
datafile = '2014-03-26-000-Data.csv'

date, time, millis, ax, ay, az, rollrate, pitchrate, yawrate, roll, pitch, yaw, speed, course, latitude, longitude, altitude, pdop, hdop, vdop, epe, fix, satellites_view, satellites_used, temp = np.loadtxt(datafile, delimiter=',', unpack=True, 
                  #converters={1: mdates.strpdate2num('%H%M%S%f'),
                  #            0: mdates.strpdate2num('%y%m%d')},
                  skiprows=1)

print('Read \'%s\' successfully.' % datafile)

# A course of 0° means the Car is traveling north bound
# and 90° means it is traveling east bound.
# In the Calculation following, East is Zero and North is 90°
# We need an offset.
course =(-course+90.0)


# ## Measurement Function H

# Matrix $J_H$ is the Jacobian of the Measurement function $h$ with respect to the state. Function $h$ can be used to compute the predicted measurement from the predicted state.
# 
# If a GPS measurement is available, the following function maps the state to the measurement.

# In[34]:


hs = Matrix([[xs],
             [ys],
             [vs],
             [dpsis],
             [axs]])
hs


# In[35]:


JHs=hs.jacobian(state)
JHs


# If no GPS measurement is available, simply set the corresponding values in $J_h$ to zero.

# ## Measurement Noise Covariance $R$

# "In practical use, the uncertainty estimates take on the significance of relative weights of state estimates and measurements. So it is not so much important that uncertainty is absolutely correct as it is that it be relatively consistent across all models" - Kelly, A. (1994). A 3D state space formulation of a navigation Kalman filter for autonomous vehicles, (May). Retrieved from http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA282853

# In[36]:


varGPS = 5.0 # Standard Deviation of GPS Measurement
varspeed = 3.0 # Variance of the speed measurement
varyaw = 0.1 # Variance of the yawrate measurement
varacc = 1.0 # Variance of the longitudinal Acceleration
R = np.diag([varGPS**2, varGPS**2, varspeed**2, varyaw**2, varacc**2])

print(R, R.shape)


# In[37]:


fig = plt.figure(figsize=(4.5, 4.5))
im = plt.imshow(R, interpolation="none", cmap=plt.get_cmap('binary'))
plt.title('Measurement Noise Covariance Matrix $R$')
ylocs, ylabels = plt.yticks()
# set the locations of the yticks
plt.yticks(np.arange(6))
# set the locations and labels of the yticks
plt.yticks(np.arange(5),('$x$', '$y$', '$v$', '$\dot \psi$', '$a$'), fontsize=22)

xlocs, xlabels = plt.xticks()
# set the locations of the yticks
plt.xticks(np.arange(6))
# set the locations and labels of the yticks
plt.xticks(np.arange(5),('$x$', '$y$', '$v$', '$\dot \psi$', '$a$'), fontsize=22)

plt.xlim([-0.5,4.5])
plt.ylim([4.5, -0.5])

from mpl_toolkits.axes_grid1 import make_axes_locatable
divider = make_axes_locatable(plt.gca())
cax = divider.append_axes("right", "5%", pad="3%")
plt.colorbar(im, cax=cax);


# ## Identity Matrix

# In[38]:


I = np.eye(numstates)
print(I, I.shape)


# ## Approx. Lat/Lon to Meters to check Location

# In[39]:


RadiusEarth = 6378388.0 # m
arc= 2.0*np.pi*(RadiusEarth+altitude)/360.0 # m/°

dx = arc * np.cos(latitude*np.pi/180.0) * np.hstack((0.0, np.diff(longitude))) # in m
dy = arc * np.hstack((0.0, np.diff(latitude))) # in m

mx = np.cumsum(dx)
my = np.cumsum(dy)

ds = np.sqrt(dx**2+dy**2)

GPS=(ds!=0.0).astype('bool') # GPS Trigger for Kalman Filter


# ## Initial State

# In[40]:


x = np.matrix([[mx[0], my[0], course[0]/180.0*np.pi, speed[0]/3.6+0.001, yawrate[0]/180.0*np.pi, ax[0]]]).T
print(x, x.shape)

U=float(np.cos(x[2])*x[3])
V=float(np.sin(x[2])*x[3])

plt.quiver(x[0], x[1], U, V)
plt.scatter(float(x[0]), float(x[1]), s=100)
plt.title('Initial Location')
plt.axis('equal')


# ### Put everything together as a measurement vector

# In[41]:


measurements = np.vstack((mx, my, speed/3.6, yawrate/180.0*np.pi, ax))
# Lenth of the measurement
m = measurements.shape[1]
print(measurements.shape)


# In[42]:


# Preallocation for Plotting
x0 = []
x1 = []
x2 = []
x3 = []
x4 = []
x5 = []
Zx = []
Zy = []
Px = []
Py = []
Pdx= []
Pdy= []
Pddx=[]
Pddy=[]
Pdv =[]
Kx = []
Ky = []
Kdx= []
Kdy= []
Kddx=[]
Kdv= []
dstate=[]


# # Extended Kalman Filter

# ![Extended Kalman Filter Step](Extended-Kalman-Filter-Step.png)

# $$x_k= \begin{bmatrix} x \\ y \\ \psi \\ v \\ \dot\psi \\ a \end{bmatrix} = \begin{bmatrix} \text{Position X} \\ \text{Position Y} \\ \text{Heading} \\ \text{Velocity} \\ \text{Yaw Rate} \\ \text{acceleration} \end{bmatrix} =  \underbrace{\begin{matrix}x[0] \\ x[1] \\ x[2] \\ x[3] \\ x[4] \\ x[5]  \end{matrix}}_{\textrm{Python Nomenclature}}$$

# In[43]:


for filterstep in range(m):

    # Time Update (Prediction)
    # ========================
    # Project the state ahead
    # see "Dynamic Matrix"
    if np.abs(yawrate[filterstep])<0.0001: # Driving straight
        x[4] = 0.0001
    x[0] = x[0] + (1 / x[4]**2) * ((x[3]*x[4] + x[5] * x[4] * dt) *         np.sin(x[2] + x[4]* dt) + x[5] * np.cos(x[2] + x[4] * dt) - x[3] *          x[4] * np.sin(x[2]) - x[5] * np.cos(x[2]))
    x[1] = x[1] + (1 / x[4]**2) * ((-x[3]*x[4] - x[5] * x[4] * dt) *         np.cos(x[2] + x[4]* dt) + x[5] * np.sin(x[2] + x[4] * dt) + x[3] *         x[4] * np.cos(x[2]) - x[5] * np.sin(x[2]))
    x[2] = (x[2] + x[4] * dt + np.pi) % (2.0 * np.pi) - np.pi
    x[3] = x[3] + x[5] * dt 
    x[4] = x[4]
    x[5] = x[5]
    
    
    # Calculate the Jacobian of the Dynamic Matrix A
    # see "Calculate the Jacobian of the Dynamic Matrix with respect to the state vector"
    a13 = ((-x[4]*x[3]*np.cos(x[2]) + x[5]*np.sin(x[2]) - x[5]*np.sin(dt*x[4] + x[2]) +         (dt*x[4]*x[5] + x[4]*x[3])*np.cos(dt*x[4] + x[2]))/x[4]**2).item(0)

    a14 = ((-x[4]*np.sin(x[2]) + x[4]*np.sin(dt*x[4] + x[2]))/x[4]**2).item(0)

    a15 = ((-dt*x[5]*np.sin(dt*x[4] + x[2]) + dt*(dt*x[4]*x[5] + x[4]*x[3])*         np.cos(dt*x[4] + x[2]) - x[3]*np.sin(x[2]) + (dt*x[5] + x[3])*         np.sin(dt*x[4] + x[2]))/x[4]**2 - 2*(-x[4]*x[3]*np.sin(x[2]) - x[5]*         np.cos(x[2]) + x[5]*np.cos(dt*x[4] + x[2]) + (dt*x[4]*x[5] + x[4]*x[3])*         np.sin(dt*x[4] + x[2]))/x[4]**3).item(0)

    a16 = ((dt*x[4]*np.sin(dt*x[4] + x[2]) - np.cos(x[2]) + np.cos(dt * x[4] + x[2]))/x[4]**2).item(0)

    a23 = ((-x[4] * x[3] * np.sin(x[2]) - x[5] * np.cos(x[2]) + x[5] * np.cos(dt * x[4] + x[2]) -         (-dt * x[4]*x[5] - x[4] * x[3]) * np.sin(dt * x[4] + x[2])) / x[4]**2).item(0)
    a24 = ((x[4] * np.cos(x[2]) - x[4]*np.cos(dt*x[4] + x[2]))/x[4]**2).item(0)
    a25 = ((dt * x[5]*np.cos(dt*x[4] + x[2]) - dt * (-dt*x[4]*x[5] - x[4] * x[3]) *         np.sin(dt * x[4] + x[2]) + x[3]*np.cos(x[2]) + (-dt*x[5] - x[3])*np.cos(dt*x[4] + x[2]))/         x[4]**2 - 2*(x[4]*x[3]*np.cos(x[2]) - x[5] * np.sin(x[2]) + x[5] * np.sin(dt*x[4] + x[2]) +         (-dt * x[4] * x[5] - x[4] * x[3])*np.cos(dt*x[4] + x[2]))/x[4]**3).item(0)
    a26 =  ((-dt*x[4]*np.cos(dt*x[4] + x[2]) - np.sin(x[2]) + np.sin(dt*x[4] + x[2]))/x[4]**2).item(0)
        
    JA = np.matrix([[1.0, 0.0, a13, a14, a15, a16],
                    [0.0, 1.0, a23, a24, a25, a26],
                    [0.0, 0.0, 1.0, 0.0, dt, 0.0],
                    [0.0, 0.0, 0.0, 1.0, 0.0, dt],
                    [0.0, 0.0, 0.0, 0.0, 1.0, 0.0],
                    [0.0, 0.0, 0.0, 0.0, 0.0, 1.0]])
    
    
    # Project the error covariance ahead
    P = JA*P*JA.T + Q
    
    # Measurement Update (Correction)
    # ===============================
    # Measurement Function
    hx = np.matrix([[float(x[0])],
                    [float(x[1])],
                    [float(x[3])],
                    [float(x[4])],
                    [float(x[5])]])

    if GPS[filterstep]: # with 10Hz, every 5th step
        JH = np.matrix([[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, 0.0, 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]])
    else: # every other step
        JH = np.matrix([[0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
                        [0.0, 0.0, 0.0, 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, 0.0, 0.0, 0.0, 1.0]])        
    
    S = JH*P*JH.T + R
    K = (P*JH.T) * np.linalg.inv(S)

    # Update the estimate via
    Z = measurements[:,filterstep].reshape(JH.shape[0],1)
    y = Z - (hx)                         # Innovation or Residual
    x = x + (K*y)

    # Update the error covariance
    P = (I - (K*JH))*P


    # Save states for Plotting
    x0.append(float(x[0]))
    x1.append(float(x[1]))
    x2.append(float(x[2]))
    x3.append(float(x[3]))
    x4.append(float(x[4]))
    x5.append(float(x[5]))
    Zx.append(float(Z[0]))
    Zy.append(float(Z[1]))    
    Px.append(float(P[0,0]))
    Py.append(float(P[1,1]))
    Pdx.append(float(P[2,2]))
    Pdy.append(float(P[3,3]))
    Pddx.append(float(P[4,4]))
    Pdv.append(float(P[5,5]))
    Kx.append(float(K[0,0]))
    Ky.append(float(K[1,0]))
    Kdx.append(float(K[2,0]))
    Kdy.append(float(K[3,0]))
    Kddx.append(float(K[4,0]))
    Kdv.append(float(K[5,0]))


# ## Plots

# ### Uncertainties

# In[44]:


fig = plt.figure(figsize=(16,9))
plt.semilogy(range(m),Px, label='$x$')
plt.step(range(m),Py, label='$y$')
plt.step(range(m),Pdx, label='$\psi$')
plt.step(range(m),Pdy, label='$v$')
plt.step(range(m),Pddx, label='$\dot \psi$')

plt.xlabel('Filter Step')
plt.ylabel('')
plt.title('Uncertainty (Elements from Matrix $P$)')
plt.legend(loc='best',prop={'size':22})


# In[45]:


fig = plt.figure(figsize=(6, 6))
im = plt.imshow(P, interpolation="none", cmap=plt.get_cmap('binary'))
plt.title('Covariance Matrix $P$ (after %i Filter Steps)' % (m))
ylocs, ylabels = plt.yticks()
# set the locations of the yticks
plt.yticks(np.arange(6))
# set the locations and labels of the yticks
plt.yticks(np.arange(5),('$x$', '$y$', '$\psi$', '$v$', '$\dot \psi$'), fontsize=22)

xlocs, xlabels = plt.xticks()
# set the locations of the yticks
plt.xticks(np.arange(6))
# set the locations and labels of the yticks
plt.xticks(np.arange(5),('$x$', '$y$', '$\psi$', '$v$', '$\dot \psi$'), fontsize=22)

plt.xlim([-0.5,4.5])
plt.ylim([4.5, -0.5])

from mpl_toolkits.axes_grid1 import make_axes_locatable
divider = make_axes_locatable(plt.gca())
cax = divider.append_axes("right", "5%", pad="3%")
plt.colorbar(im, cax=cax)


plt.tight_layout()


# ### Kalman Gains

# In[46]:


fig = plt.figure(figsize=(16,9))
plt.step(range(len(measurements[0])),Kx, label='$x$')
plt.step(range(len(measurements[0])),Ky, label='$y$')
plt.step(range(len(measurements[0])),Kdx, label='$\psi$')
plt.step(range(len(measurements[0])),Kdy, label='$v$')
plt.step(range(len(measurements[0])),Kddx, label='$\dot \psi$')


plt.xlabel('Filter Step')
plt.ylabel('')
plt.title('Kalman Gain (the lower, the more the measurement fullfill the prediction)')
plt.legend(prop={'size':18})
plt.ylim([-0.2,0.2]);


# ## State Vector

# In[47]:


fig = plt.figure(figsize=(16,16))

plt.subplot(511)
plt.step(range(len(measurements[0])),x0-mx[0], label='$x$')
plt.step(range(len(measurements[0])),x1-my[0], label='$y$')

plt.title('Extended Kalman Filter State Estimates (State Vector $x$)')
plt.legend(loc='best')
plt.ylabel('Position (relative to start) [m]')

plt.subplot(512)
plt.step(range(len(measurements[0])),x2, label='$\psi$')
plt.step(range(len(measurements[0])),(course/180.0*np.pi+np.pi)%(2.0*np.pi) - np.pi, label='$\psi$ (from GPS as reference)')
plt.ylabel('Course')
plt.legend(loc='best')
           
plt.subplot(513)
plt.step(range(len(measurements[0])),x3, label='$v$')
plt.step(range(len(measurements[0])),speed/3.6, label='$v$ (from GPS as reference)', alpha=0.6)
plt.ylabel('Velocity')
plt.ylim([0, 30])
plt.legend(loc='best')

plt.subplot(514)
plt.step(range(len(measurements[0])),x4, label='$\dot \psi$')
plt.step(range(len(measurements[0])),yawrate/180.0*np.pi, label='$\dot \psi$ (from IMU as reference)', alpha=0.6)
plt.ylabel('Yaw Rate')
plt.ylim([-0.6, 0.6])
plt.legend(loc='best')

plt.subplot(515)
plt.step(range(len(measurements[0])),x5, label='$a$')
plt.step(range(len(measurements[0])),ax, label='$a$ (from IMU as reference)', alpha=0.6)
plt.ylabel('Acceleration')
#plt.ylim([-0.6, 0.6])
plt.legend(loc='best')
plt.xlabel('Filter Step')

plt.savefig('Extended-Kalman-Filter-CTRA-State-Estimates.png', dpi=72, transparent=True, bbox_inches='tight')


# ## Position x/y

# In[48]:


fig = plt.figure(figsize=(16,9))

# EKF State
plt.quiver(x0,x1,np.cos(x2), np.sin(x2), color='#94C600', units='xy', width=0.05, scale=0.5)
plt.plot(x0,x1, label='EKF Position', c='k', lw=5)

# Measurements
plt.scatter(mx[::5],my[::5], s=50, label='GPS Measurements', marker='+')
#cbar=plt.colorbar(ticks=np.arange(20))
#cbar.ax.set_ylabel(u'EPE', rotation=270)
#cbar.ax.set_xlabel(u'm')

# Start/Goal
plt.scatter(x0[0],x1[0], s=60, label='Start', c='g')
plt.scatter(x0[-1],x1[-1], s=60, label='Goal', c='r')

plt.xlabel('X [m]')
plt.ylabel('Y [m]')
plt.title('Position')
plt.legend(loc='best')
plt.axis('equal')
#plt.tight_layout()

#plt.savefig('Extended-Kalman-Filter-CTRA-Position.png', dpi=72, transparent=True, bbox_inches='tight')


# ### Detailed View

# In[49]:


fig = plt.figure(figsize=(12,9))

plt.subplot(221)
# EKF State
#plt.quiver(x0,x1,np.cos(x2), np.sin(x2), color='#94C600', units='xy', width=0.05, scale=0.5)
plt.plot(x0,x1, label='EKF Position', c='g', lw=5)

# Measurements
plt.scatter(mx[::5],my[::5], s=50, label='GPS Measurements', alpha=0.5, marker='+')
#cbar=plt.colorbar(ticks=np.arange(20))
#cbar.ax.set_ylabel(u'EPE', rotation=270)
#cbar.ax.set_xlabel(u'm')

plt.xlabel('X [m]')
plt.xlim(70, 130)
plt.ylabel('Y [m]')
plt.ylim(140, 200)
plt.title('Position')
plt.legend(loc='best')


plt.subplot(222)

# EKF State
#plt.quiver(x0,x1,np.cos(x2), np.sin(x2), color='#94C600', units='xy', width=0.05, scale=0.5)
plt.plot(x0,x1, label='EKF Position', c='g', lw=5)

# Measurements
plt.scatter(mx[::5],my[::5], s=50, label='GPS Measurements', alpha=0.5, marker='+')
#cbar=plt.colorbar(ticks=np.arange(20))
#cbar.ax.set_ylabel(u'EPE', rotation=270)
#cbar.ax.set_xlabel(u'm')

plt.xlabel('X [m]')
plt.xlim(160, 260)
plt.ylabel('Y [m]')
plt.ylim(110, 160)
plt.title('Position')
plt.legend(loc='best')


# # Conclusion

# As you can see, complicated analytic calculation of the Jacobian Matrices, but it works pretty well.

# ## Write Google Earth KML

# ### Convert back from Meters to Lat/Lon (WGS84)

# In[30]:


latekf = latitude[0] + np.divide(x1,arc)
lonekf = longitude[0]+ np.divide(x0,np.multiply(arc,np.cos(latitude*np.pi/180.0)))


# ### Create Data for KML Path

# Coordinates and timestamps to be used to locate the car model in time and space
# The value can be expressed as yyyy-mm-ddThh:mm:sszzzzzz, where T is the separator between the date and the time, and the time zone is either Z (for UTC) or zzzzzz, which represents ±hh:mm in relation to UTC.

# In[31]:


import datetime
car={}
car['when']=[]
car['coord']=[]
car['gps']=[]
for i in range(len(millis)):
    d=datetime.datetime.fromtimestamp(millis[i]/1000.0)
    car["when"].append(d.strftime("%Y-%m-%dT%H:%M:%SZ"))
    car["coord"].append((lonekf[i], latekf[i], 0))
    car["gps"].append((longitude[i], latitude[i], 0))


# In[32]:


from simplekml import Kml, Model, AltitudeMode, Orientation, Scale, Style, Color


# In[33]:


# The model path and scale variables
car_dae = r'https://raw.githubusercontent.com/balzer82/Kalman/master/car-model.dae'
car_scale = 1.0

# Create the KML document
kml = Kml(name=d.strftime("%Y-%m-%d %H:%M"), open=1)

# Create the model
model_car = Model(altitudemode=AltitudeMode.clamptoground,
                            orientation=Orientation(heading=75.0),
                            scale=Scale(x=car_scale, y=car_scale, z=car_scale))

# Create the track
trk = kml.newgxtrack(name="EKF", altitudemode=AltitudeMode.clamptoground,
                     description="State Estimation from Extended Kalman Filter with CTRA Model")

# Attach the model to the track
trk.model = model_car
trk.model.link.href = car_dae

# Add all the information to the track
trk.newwhen(car["when"])
trk.newgxcoord(car["coord"])

# Style of the Track
trk.iconstyle.icon.href = ""
trk.labelstyle.scale = 1
trk.linestyle.width = 4
trk.linestyle.color = '7fff0000'

# Add GPS measurement marker
fol = kml.newfolder(name="GPS Measurements")
sharedstyle = Style()
sharedstyle.iconstyle.icon.href = 'http://maps.google.com/mapfiles/kml/shapes/placemark_circle.png'

for m in range(len(latitude)):
    if GPS[m]:
        pnt = fol.newpoint(coords = [(longitude[m],latitude[m])])
        pnt.style = sharedstyle

# Saving
kml.savekmz("Extended-Kalman-Filter-CTRA.kmz")
print('Exported KMZ File for Google Earth')