基于Python实现粒子滤波效果

1、建立仿真模型

（1）假设有一辆小车在一平面运动，起始坐标为[0，0]，运动速度为1m/s，加速度为0.1 m / s 2 m/s^2 m/s2，则可以建立如下的状态方程：
Y = A ∗ X + B ∗ U Y=A*X+B*U Y=A∗X+B∗U
U为速度和加速度的的矩阵
U = [ 1 0.1 ] U= \begin{bmatrix} 1 \\ 0.1\\ \end{bmatrix} U=[10.1​]
X为当前时刻的坐标，速度，加速度
X = [ x y y a w V ] X= \begin{bmatrix} x \\ y \\ yaw \\ V \end{bmatrix} X=⎣⎢⎢⎡​xyyawV​⎦⎥⎥⎤​
Y为下一时刻的状态
则观察矩阵A为：
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=⎣⎢⎢⎡​1000​0100​0010​0000​⎦⎥⎥⎤​
矩阵B则决定小车的运动规矩，这里取B为：
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)∗t01​00t0​⎦⎥⎥⎤​
python编程实现小车的运动轨迹:

"""

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, 0],
         [0, 1.0, 0, 0],
         [0, 0, 1.0, 0],
         [0, 0, 0, 0]])

  B = np.array([[DT * math.cos(x[2, 0]), 0],
         [DT * math.sin(x[2, 0]), 0],
         [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()

运行结果：

2、生成观测数据

实际运用中，我们需要对小车的位置进行定位，假设坐标系上有4个观测点，在小车运动过程中，需要定时将小车距离这4个观测点的位置距离记录下来，这样，当小车下一次寻迹时就有了参考点；

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, 0] - rf_id[i, 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, 0], 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, xd, ud

3、实现粒子滤波

#
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, z, u):
  """
  Localization with Particle filter
  """

  for ip in range(NP):
    x = np.array([px[:, ip]]).T
    w = pw[0, ip]

    # 预测输入
    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)

    # 计算权重
    for i in range(len(z[:, 0])):
      dx = x[0, 0] - z[i, 1]
      dy = x[1, 0] - z[i, 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() # 归一化

  x_est = px.dot(pw.T)
  p_est = calc_covariance(x_est, px, pw)
  #计算有效粒子数
  N_eff = 1.0 / (pw.dot(pw.T))[0, 0] 
  #重采样
  if N_eff < NTh:
    px, pw = re_sampling(px, pw)
  return x_est, p_est, px, pw


def re_sampling(px, pw):
  """
  low variance re-sampling
  """

  w_cum = np.cumsum(pw)
  base = np.arange(0.0, 1.0, 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、完整源码

该代码来源于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, 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, 0] - rf_id[i, 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, 0], 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, xd, ud


def motion_model(x, u):
  F = np.array([[1.0, 0, 0, 0],
         [0, 1.0, 0, 0],
         [0, 0, 1.0, 0],
         [0, 0, 0, 0]])

  B = np.array([[DT * math.cos(x[2, 0]), 0],
         [DT * math.sin(x[2, 0]), 0],
         [0.0, DT],
         [1.0, 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, px, 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, pw, z, u):
  """
  Localization with Particle filter
  """

  for ip in range(NP):
    x = np.array([px[:, ip]]).T
    w = pw[0, ip]

    # Predict with random input sampling
    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)

    # Calc Importance Weight
    for i in range(len(z[:, 0])):
      dx = x[0, 0] - z[i, 1]
      dy = x[1, 0] - z[i, 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() # normalize

  x_est = px.dot(pw.T)
  p_est = calc_covariance(x_est, px, pw)

  N_eff = 1.0 / (pw.dot(pw.T))[0, 0] # Effective particle number
  if N_eff < NTh:
    px, pw = re_sampling(px, pw)
  return x_est, p_est, px, pw


def re_sampling(px, pw):
  """
  low variance re-sampling
  """

  w_cum = np.cumsum(pw)
  base = np.arange(0.0, 1.0, 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


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],
           [0.0, 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, z, x_dr, ud = observation(x_true, x_dr, u, rf_id)

    x_est, PEst, px, pw = pf_localization(px, pw, z, 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, 0], z[i, 1]], [x_true[1, 0], z[i, 2]], "-k")
      plt.plot(rf_id[:, 0], 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, :]).flatten(), "-b")
      plt.plot(np.array(h_x_dr[0, :]).flatten(),
           np.array(h_x_dr[1, :]).flatten(), "-k")
      plt.plot(np.array(h_x_est[0, :]).flatten(),
           np.array(h_x_est[1, :]).flatten(), "-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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