基於Python實現粒子濾波效果
阿新 • • 發佈：2020-12-02
1、建立模擬模型
（1）假設有一輛小車在一平面運動，起始座標為[0，0]，運動速度為1m/s，加速度為0.1 m / s 2 m/s^2 m/s2，則可以建立如下的狀態方程：
Y = A ∗ X + B ∗ U Y=A*X+B*U Y=A∗X+B∗U
U為速度和加速度的的矩陣
U = [ 1 0.1 ] U= \begin{bmatrix} 1 \\ 0.1\\ \end{bmatrix} U=[10.1]
X為當前時刻的座標，速度，加速度
X = [ x y y a w V ] X= \begin{bmatrix} x \\ y \\ yaw \\ V \end{bmatrix} X=⎣⎢⎢⎡xyyawV⎦⎥⎥⎤
則觀察矩陣A為：
A = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 ] A= \begin{bmatrix} 1&0 & 0 &0 \\ 0 & 1 & 0&0 \\ 0 & 0 &1 &0 \\ 0&0 & 0 &0 \end{bmatrix} A=⎣⎢⎢⎡1000010000100000⎦⎥⎥⎤
矩陣B則決定小車的運動規矩，這裡取B為：
B = [ c o s ( x ) ∗ t 0 s i n ( x ) ∗ t 0 0 t 1 0 ] B= \begin{bmatrix} cos(x)*t &0\\ sin(x)*t &0\\ 0&t\\ 1&0 \end{bmatrix} B=⎣⎢⎢⎡cos(x)∗tsin(x)∗t0100t0⎦⎥⎥⎤
"""
Particle Filter localization sample
author: Atsushi Sakai (@Atsushi_twi)
"""
import math
import matplotlib.pyplot as plt
import numpy as np
from scipy.spatial.transform import Rotation as Rot
DT = 0.1 # time tick [s]
SIM_TIME = 50.0 # simulation time [s]
MAX_RANGE = 20.0 # maximum observation range
# Particle filter parameter
NP = 100 # Number of Particle
NTh = NP / 2.0 # Number of particle for re-sampling
def calc_input():
v = 1.0 # [m/s]
yaw_rate = 0.1 # [rad/s]
u = np.array([[v,yaw_rate]]).T
return u
def motion_model(x,u):
F = np.array([[1.0,0],[0,1.0,0]])
B = np.array([[DT * math.cos(x[2,0]),[DT * math.sin(x[2,[0.0,DT],[1.0,0.0]])
x = F.dot(x) + B.dot(u)
return x
def main():
print(__file__ + " start!!")
time = 0.0
# State Vector [x y yaw v]'
x_true = np.zeros((4,1))
x = []
y = []
while SIM_TIME >= time:
time += DT
u = calc_input()
x_true = motion_model(x_true,u)
x.append(x_true[0])
y.append(x_true[1])
plt.plot(x,y,"-b")
if __name__ == '__main__':
main()執行結果：
2、生成觀測資料
實際運用中，我們需要對小車的位置進行定位，假設座標系上有4個觀測點，在小車運動過程中，需要定時將小車距離這4個觀測點的位置距離記錄下來，這樣，當小車下一次尋跡時就有了參考點；
def observation(x_true,xd,u,rf_id):
x_true = motion_model(x_true,u)
# add noise to gps x-y
z = np.zeros((0,3))
for i in range(len(rf_id[:,0])):
dx = x_true[0,0] - rf_id[i,0]
dy = x_true[1,1]
d = math.hypot(dx,dy)
if d <= MAX_RANGE:
dn = d + np.random.randn() * Q_sim[0,0] ** 0.5 # add noise
zi = np.array([[dn,rf_id[i,1]]])
z = np.vstack((z,zi))
# add noise to input
ud1 = u[0,0] + np.random.randn() * R_sim[0,0] ** 0.5
ud2 = u[1,0] + np.random.randn() * R_sim[1,1] ** 0.5
ud = np.array([[ud1,ud2]]).T
xd = motion_model(xd,ud)
return x_true,z,ud
3、實現粒子濾波
#
def gauss_likelihood(x,sigma):
p = 1.0 / math.sqrt(2.0 * math.pi * sigma ** 2) * \
math.exp(-x ** 2 / (2 * sigma ** 2))
return p
def pf_localization(px,pw,u):
"""
Localization with Particle filter
"""
for ip in range(NP):
x = np.array([px[:,ip]]).T
w = pw[0,ip]
# 預測輸入
ud1 = u[0,0] + np.random.randn() * R[0,0] ** 0.5
ud2 = u[1,0] + np.random.randn() * R[1,1] ** 0.5
ud = np.array([[ud1,ud2]]).T
x = motion_model(x,ud)
# 計算權重
for i in range(len(z[:,0])):
dx = x[0,0] - z[i,1]
dy = x[1,2]
pre_z = math.hypot(dx,dy)
dz = pre_z - z[i,0]
w = w * gauss_likelihood(dz,math.sqrt(Q[0,0]))
px[:,ip] = x[:,0]
pw[0,ip] = w
pw = pw / pw.sum() # 歸一化
x_est = px.dot(pw.T)
p_est = calc_covariance(x_est,px,pw)
#計算有效粒子數
N_eff = 1.0 / (pw.dot(pw.T))[0,0]
#重取樣
if N_eff < NTh:
px,pw = re_sampling(px,pw)
return x_est,p_est,pw
def re_sampling(px,pw):
"""
low variance re-sampling
"""
w_cum = np.cumsum(pw)
base = np.arange(0.0,1 / NP)
re_sample_id = base + np.random.uniform(0,1 / NP)
indexes = []
ind = 0
for ip in range(NP):
while re_sample_id[ip] > w_cum[ind]:
ind += 1
indexes.append(ind)
px = px[:,indexes]
pw = np.zeros((1,NP)) + 1.0 / NP # init weight
return px,pw
4、完整原始碼
該程式碼來源於https://github.com/AtsushiSakai/PythonRobotics
"""
Particle Filter localization sample
author: Atsushi Sakai (@Atsushi_twi)
"""
import math
import matplotlib.pyplot as plt
import numpy as np
from scipy.spatial.transform import Rotation as Rot
# Estimation parameter of PF
Q = np.diag([0.2]) ** 2 # range error
R = np.diag([2.0,np.deg2rad(40.0)]) ** 2 # input error
# Simulation parameter
Q_sim = np.diag([0.2]) ** 2
R_sim = np.diag([1.0,np.deg2rad(30.0)]) ** 2
DT = 0.1 # time tick [s]
SIM_TIME = 50.0 # simulation time [s]
MAX_RANGE = 20.0 # maximum observation range
# Particle filter parameter
NP = 100 # Number of Particle
NTh = NP / 2.0 # Number of particle for re-sampling
show_animation = True
def calc_input():
v = 1.0 # [m/s]
yaw_rate = 0.1 # [rad/s]
u = np.array([[v,yaw_rate]]).T
return u
def observation(x_true,ud
def motion_model(x,0.0]])
x = F.dot(x) + B.dot(u)
return x
def gauss_likelihood(x,sigma):
p = 1.0 / math.sqrt(2.0 * math.pi * sigma ** 2) * \
math.exp(-x ** 2 / (2 * sigma ** 2))
return p
def calc_covariance(x_est,pw):
"""
calculate covariance matrix
see ipynb doc
"""
cov = np.zeros((3,3))
n_particle = px.shape[1]
for i in range(n_particle):
dx = (px[:,i:i + 1] - x_est)[0:3]
cov += pw[0,i] * dx @ dx.T
cov *= 1.0 / (1.0 - pw @ pw.T)
return cov
def pf_localization(px,ip]
# Predict with random input sampling
ud1 = u[0,ud)
# Calc Importance Weight
for i in range(len(z[:,ip] = w
pw = pw / pw.sum() # normalize
x_est = px.dot(pw.T)
p_est = calc_covariance(x_est,pw)
N_eff = 1.0 / (pw.dot(pw.T))[0,0] # Effective particle number
if N_eff < NTh:
px,pw
def plot_covariance_ellipse(x_est,p_est): # pragma: no cover
p_xy = p_est[0:2,0:2]
eig_val,eig_vec = np.linalg.eig(p_xy)
if eig_val[0] >= eig_val[1]:
big_ind = 0
small_ind = 1
else:
big_ind = 1
small_ind = 0
t = np.arange(0,2 * math.pi + 0.1,0.1)
# eig_val[big_ind] or eiq_val[small_ind] were occasionally negative
# numbers extremely close to 0 (~10^-20),catch these cases and set the
# respective variable to 0
try:
a = math.sqrt(eig_val[big_ind])
except ValueError:
a = 0
try:
b = math.sqrt(eig_val[small_ind])
except ValueError:
b = 0
x = [a * math.cos(it) for it in t]
y = [b * math.sin(it) for it in t]
angle = math.atan2(eig_vec[1,big_ind],eig_vec[0,big_ind])
rot = Rot.from_euler('z',angle).as_matrix()[0:2,0:2]
fx = rot.dot(np.array([[x,y]]))
px = np.array(fx[0,:] + x_est[0,0]).flatten()
py = np.array(fx[1,:] + x_est[1,0]).flatten()
plt.plot(px,py,"--r")
def main():
print(__file__ + " start!!")
time = 0.0
# RF_ID positions [x,y]
rf_id = np.array([[10.0,0.0],[10.0,10.0],15.0],[-5.0,20.0]])
# State Vector [x y yaw v]'
x_est = np.zeros((4,1))
x_true = np.zeros((4,1))
px = np.zeros((4,NP)) # Particle store
pw = np.zeros((1,NP)) + 1.0 / NP # Particle weight
x_dr = np.zeros((4,1)) # Dead reckoning
# history
h_x_est = x_est
h_x_true = x_true
h_x_dr = x_true
while SIM_TIME >= time:
time += DT
u = calc_input()
x_true,x_dr,ud = observation(x_true,rf_id)
x_est,PEst,pw = pf_localization(px,ud)
# store data history
h_x_est = np.hstack((h_x_est,x_est))
h_x_dr = np.hstack((h_x_dr,x_dr))
h_x_true = np.hstack((h_x_true,x_true))
if show_animation:
plt.cla()
# for stopping simulation with the esc key.
plt.gcf().canvas.mpl_connect(
'key_release_event',lambda event: [exit(0) if event.key == 'escape' else None])
for i in range(len(z[:,0])):
plt.plot([x_true[0,z[i,1]],[x_true[1,2]],"-k")
plt.plot(rf_id[:,rf_id[:,1],"*k")
plt.plot(px[0,:],px[1,".r")
plt.plot(np.array(h_x_true[0,:]).flatten(),np.array(h_x_true[1,"-b")
plt.plot(np.array(h_x_dr[0,np.array(h_x_dr[1,"-k")
plt.plot(np.array(h_x_est[0,np.array(h_x_est[1,"-r")
plot_covariance_ellipse(x_est,PEst)
plt.axis("equal")
plt.grid(True)
plt.pause(0.001)
if __name__ == '__main__':
main()
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