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utils_ekf.py
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utils_ekf.py
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'''
This script contains functions to reconstruct the ego vehicle and
surrounding vehicles using Extended Kalman Filter (EKF).
'''
import numpy as np
# Reconstruct the trajectory of the ego/subject vehicle
# Extended Kalman Filter for Constant Heading and Acceleration,
# adapted from https://github.com/balzer82/Kalman/blob/master/Extended-Kalman-Filter-CTRA.ipynb
def reconstruct_ego(df_ego, params=[], reverse=False):
if len(params)==0:
uncertainty_init=100.
uncertainty_speed=100.
uncertainty_omega=10.
uncertainty_acc=10.
max_jerk=0.5
max_yaw_rate=0.1
max_acc=9.8
max_yaw_acc=1.
else:
uncertainty_init, uncertainty_speed, uncertainty_omega, uncertainty_acc, max_jerk, max_yaw_rate, max_acc, max_yaw_acc = params
veh = df_ego.sort_values('time').copy().reset_index(drop=True)
if reverse:
veh = veh.iloc[::-1].reset_index(drop=True)
## Constants
g = 9.81 ### gravity, m/s^2
mph2mps = 0.44704 ### mph to m/s
## Convert yaw rate to radians per second
veh['yaw_rate'] = np.deg2rad(veh['yaw_rate'])
## Convert acceleration to m/s^2
veh['acc_lat'] = veh['acc_lat'] * g
veh['acc_lon'] = veh['acc_lon'] * g
## Convert speed to m/s
veh['speed_comp'] = veh['speed_comp']*mph2mps
## Initialize
numstates = 6
P = np.eye(numstates)*uncertainty_init # Initial Uncertainty
R = np.diag([uncertainty_speed,uncertainty_omega,uncertainty_acc]) # Measurement Noise
I = np.eye(numstates)
dt = np.gradient(veh['time'])
acc_square = (veh['acc_lat']**2+veh['acc_lon']**2).values
Trigger = (acc_square>0.).astype('bool') # Perform EKF when acceleration is not zero
## Measurement vector
mv = veh['speed_comp'].values
momega = veh['yaw_rate'].values
momega[(momega<1e-6)&(momega>=0)] = 1e-6
momega[(momega>-1e-6)&(momega<0)] = -1e-6
macc = veh['acc_lon'].values
measurements = np.vstack((mv,momega,macc))
m = measurements.shape[1]
## Initial state
x = np.array([0,0,0,mv[0],momega[0],macc[0]])
## Estimated vector
estimates = np.zeros((m,numstates))
estimates[0,:] = x
for filterstep in np.arange(1,m):
## Time Update (Prediction)
x[0] = x[0] + (1/x[4]**2) * (-x[3]*x[4]*np.sin(x[2]) -x[5]*np.cos(x[2]) +
x[5]*np.cos(x[2]+x[4]*dt[filterstep]) +
(x[5]*x[4]*dt[filterstep]+x[3]*x[4])*np.sin(x[2]+x[4]*dt[filterstep]))
x[1] = x[1] + (1/x[4]**2) * (x[3]*x[4]*np.cos(x[2]) - x[5]*np.sin(x[2]) +
x[5]*np.sin(x[2]+x[4]*dt[filterstep]) +
(-x[5]*x[4]*dt[filterstep]-x[3]*x[4])*np.cos(x[2]+x[4]*dt[filterstep]))
x[2] = (x[2] + x[4] * dt[filterstep] + np.pi) % (2.0 * np.pi) - np.pi
x[3] = x[3] + x[5] * dt[filterstep]
x[4] = x[4]
x[5] = x[5]
## Calculate the Jacobian of the Dynamic Matrix JA
a13 = (-x[4]*x[3]*np.cos(x[2]) + x[5]*np.sin(x[2]) - x[5]*np.sin(dt[filterstep]*x[4]+x[2]) +
(dt[filterstep]*x[4]*x[5]+x[4]*x[3])*np.cos(dt[filterstep]*x[4]+x[2])) / x[4]**2
a14 = (-x[4]*np.sin(x[2]) + x[4]*np.sin(dt[filterstep]*x[4]+x[2])) / x[4]**2
a15 = (-dt[filterstep]*x[5]*np.sin(dt[filterstep]*x[4]+x[2]) +
dt[filterstep]*(dt[filterstep]*x[4]*x[5]+x[4]*x[3])*np.cos(dt[filterstep]*x[4]+x[2]) -
x[3]*np.sin(x[2]) + (dt[filterstep]*x[5] + x[3])*np.sin(dt[filterstep]*x[4]+x[2]))/x[4]**2 - (
-x[4]*x[3]*np.sin(x[2]) - x[5]*np.cos(x[2]) +
x[5]*np.cos(dt[filterstep]*x[4] + x[2]) +
(dt[filterstep]*x[4]*x[5] + x[4]*x[3])*np.sin(dt[filterstep]*x[4] + x[2])) *2 / x[4]**3
a16 = (dt[filterstep]*x[4]*np.sin(dt[filterstep]*x[4]+x[2]) - np.cos(x[2]) + np.cos(dt[filterstep]*x[4]+x[2])) / x[4]**2
a23 = (-x[4]*x[3]*np.sin(x[2]) - x[5]*np.cos(x[2]) + x[5]*np.cos(dt[filterstep]*x[4]+x[2]) -
(-dt[filterstep]*x[4]*x[5] - x[4]*x[3])*np.sin(dt[filterstep]*x[4]+x[2])) / x[4]**2
a24 = (x[4]*np.cos(x[2]) - x[4]*np.cos(dt[filterstep]*x[4] + x[2])) / x[4]**2
a25 = (dt[filterstep]*x[5]*np.cos(dt[filterstep]*x[4] + x[2]) -
dt[filterstep]*(-dt[filterstep]*x[4]*x[5]-x[4]*x[3])*np.sin(dt[filterstep]*x[4]+x[2]) +
x[3]*np.cos(x[2]) + (-dt[filterstep]*x[5]-x[3])*np.cos(dt[filterstep]*x[4]+x[2]))/x[4]**2 - (
x[4]*x[3]*np.cos(x[2]) - x[5]*np.sin(x[2]) +
x[5]*np.sin(dt[filterstep]*x[4]+x[2]) +
(-dt[filterstep]*x[4]*x[5]-x[4]*x[3])*np.cos(dt[filterstep]*x[4]+x[2])) *2 / x[4]**3
a26 = (-dt[filterstep]*x[4]*np.cos(dt[filterstep]*x[4]+x[2]) - np.sin(x[2]) + np.sin(dt[filterstep]*x[4] + x[2])) / x[4]**2
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[filterstep], 0.0],
[0.0, 0.0, 0.0, 1.0, 0.0, dt[filterstep]],
[0.0, 0.0, 0.0, 0.0, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 0.0, 1.0]], dtype=float)
## Calculate the Process Noise Covariance Matrix
s_pos = 0.5*max_acc*dt[filterstep]**2
s_psi = max_yaw_rate*dt[filterstep]
s_speed = max_acc*dt[filterstep]
s_omega = max_yaw_acc*dt[filterstep]
s_acc = max_jerk*dt[filterstep]
Q = np.diag([s_pos**2, s_pos**2, s_psi**2, s_speed**2, s_omega**2, s_acc**2])
## Project the error covariance ahead
P = JA*P*JA.T + Q
## Measurement Update (Correction)
hx = np.matrix([[x[3]],[x[4]],[x[5]]])
if Trigger[filterstep]:
JH = np.matrix([[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]], dtype=float)
else:
JH = np.matrix([[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, 0.0]], dtype=float)
S = JH*P*JH.T + R
K = (P*JH.T) * np.linalg.inv(S.astype('float'))
## Update the estimate
Z = measurements[:,filterstep].reshape(JH.shape[0],1)
y = Z - (hx) ### Innovation or Residual
## Check if the measurement is invalid
### 1) the speed measurement is -1
### 2) the speed measurement drops to 0 although the vehile is accelerating
if mv[filterstep]<0. or (mv[filterstep]<=0. and macc[filterstep-1:filterstep+2].mean()>0.):
y[0] = 0.
x = x + np.array(K*y).reshape(-1)
## Limit the speed to be non-negative
if x[3]<0:
x[3] = 0.
## Update the error covariance
P = (I - (K*JH))*P
## Save states
estimates[filterstep,:] = x
veh[['x_ekf','y_ekf','psi_ekf','v_ekf','omega_ekf','acc_ekf']] = estimates
if reverse:
veh = veh.iloc[::-1].reset_index(drop=True)
return veh
# Reconstruct the trajectory of the surrounding vehicles
# Extended Kalman Filter for Constant Heading and Velocity
# Adapted from https://github.com/balzer82/Kalman/blob/master/Extended-Kalman-Filter-CHCV.ipynb
def reconstruct_surrounding(veh, params=[]):
if len(params)==0:
uncertainty_init=100.
uncertainty_pos=50.
uncertainty_speed=10.
max_acc=9.8
max_yaw_rate=np.pi/2
else:
uncertainty_init, uncertainty_pos, uncertainty_speed, max_acc, max_yaw_rate = params
## Initialize
numstates = 4
P = np.eye(numstates)*uncertainty_init # Initial Uncertainty
dt = np.gradient(veh['time'])
R = np.diag([uncertainty_pos,uncertainty_pos,uncertainty_speed]) # Measurement Noise
I = np.eye(numstates)
mx, my, mv = veh['x'].values, veh['y'].values, veh['speed_comp'].values
Trigger = (mv>0.).astype('bool') # Perform EKF when speed is not zero
## Measurement vector
measurements = np.vstack((mx, my, mv))
m = measurements.shape[1]
## Initial state
x = np.array([mx[0], my[0], mv[0], 0.])
## Estimated vector
estimates = np.zeros((m,4))
for filterstep in range(m):
## Time Update (Prediction)
x[0] = x[0] + dt[filterstep]*x[2]*np.cos(x[3])
x[1] = x[1] + dt[filterstep]*x[2]*np.sin(x[3])
x[2] = x[2]
x[3] = (x[3]+ np.pi) % (2.0*np.pi) - np.pi
## Calculate the Jacobian of the Dynamic Matrix JA
a13 = dt[filterstep]*np.cos(x[3])
a14 = -dt[filterstep]*x[2]*np.sin(x[3])
a23 = dt[filterstep]*np.sin(x[3])
a24 = dt[filterstep]*x[2]*np.cos(x[3])
JA = np.matrix([[1.0, 0.0, a13, a14],
[0.0, 1.0, a23, a24],
[0.0, 0.0, 1.0, 0.0],
[0.0, 0.0, 0.0, 1.0]], dtype=float)
## Calculate the Process Noise Covariance Matrix
s_pos = 0.5*max_acc*dt[filterstep]**2
s_psi = max_yaw_rate*dt[filterstep]
s_speed = max_acc*dt[filterstep]
Q = np.diag([s_pos**2, s_pos**2, s_speed**2, s_psi**2])
## Project the error covariance ahead
P = JA*P*JA.T + Q
## Measurement Update (Correction)
hx = np.matrix([[x[0]],[x[1]],[x[2]]])
if Trigger[filterstep]:
JH = np.matrix([[1.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0]], dtype=float)
else:
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]], dtype=float)
S = JH*P*JH.T + R
K = (P*JH.T) * np.linalg.inv(S.astype('float'))
## Update the estimate
Z = measurements[:,filterstep].reshape(JH.shape[0],1)
y = Z - (hx) # Innovation or Residual
x = x + np.array(K*y).reshape(-1)
## Update the error covariance
P = (I - (K*JH))*P
## Save states
estimates[filterstep,:] = x
veh[['x_ekf','y_ekf','v_ekf','psi_ekf']] = estimates
return veh