粒子滤波（Particle Filter）及其在机器人定位中的应用
一、粒子滤波简介
粒子滤波的基本思想是用随机样本来描述概率分布，然后在测量的基础上，通过调节各粒子权值的大小和样本的位置，来近似实际概率分布，并以样本的均值作为系统的估计值。
粒子滤波是指：通过寻找一组在状态空间中传播的随机样本对概率密度函数P(Xk|Zk)进行近似，以样本均值代替积分运算，从而获得状态最小方差估计的过程，这些样本即称为“粒子。采用数学语言描述如下：对于平稳的随机过程，假定k-1时刻系统的后验概率密度为P(Xk-1|Zk-1)，依据一定原则选取n个随机样本点，k时刻获得测量信息后，经过状态和时间更新过程，n个粒子的后验概率密度可近似为P(Xk|Zk) 。随着粒子数目的增加，粒子的概率密度函数逐渐逼近状态的概率密度数，粒子滤波估计即达到了最优贝叶斯估计的效果。
粒子滤波的算法步骤如下：
粒子滤波的推导过程详见http://blog.csdn.net/heyijia0327/article/category/3245449
二、粒子滤波在机器人定位中的应用
from math import *
import random
# --------
#
# some top level parameters
#
max_steering_angle = pi / 4.0 # You do not need to use this value, but keep in mind the limitations of a real car.
bearing_noise = 0.1 # Noise parameter: should be included in sense function.
steering_noise = 0.1 # Noise parameter: should be included in move function.
distance_noise = 5.0 # Noise parameter: should be included in move function.
tolerance_xy = 15.0 # Tolerance for localization in the x and y directions.
tolerance_orientation = 0.25 # Tolerance for orientation.
# --------
#
# the "world" has 4 landmarks.
# the robot's initial coordinates are somewhere in the square
# represented by the landmarks.
#
# NOTE: Landmark coordinates are given in (y, x) form and NOT
# in the traditional (x, y) format!
landmarks = [[0.0, 100.0], [0.0, 0.0], [100.0, 0.0], [100.0, 100.0]] # position of 4 landmarks in (y, x) format.
world_size = 100.0 # world is NOT cyclic. Robot is allowed to travel "out of bounds"
# ------------------------------------------------
#
# this is the robot class
#
class robot:
# --------
# init:
# creates robot and initializes location/orientation
#
def __init__(self, length = 20.0):
self.x = random.random() * world_size # initial x position
self.y = random.random() * world_size # initial y position
self.orientation = random.random() * 2.0 * pi # initial orientation
self.length = length # length of robot
self.bearing_noise = 0.0 # initialize bearing noise to zero
self.steering_noise = 0.0 # initialize steering noise to zero
self.distance_noise = 0.0 # initialize distance noise to zero
# --------
# set:
# sets a robot coordinate
#
def set(self, new_x, new_y, new_orientation):
if new_orientation < 0 or new_orientation >= 2 * pi:
raise ValueError, 'Orientation must be in [0..2pi]'
self.x = float(new_x)
self.y = float(new_y)
self.orientation = float(new_orientation)
# --------
# set_noise:
# sets the noise parameters
#
def set_noise(self, new_b_noise, new_s_noise, new_d_noise):
# makes it possible to change the noise parameters
# this is often useful in particle filters
self.bearing_noise = float(new_b_noise)
self.steering_noise = float(new_s_noise)
self.distance_noise = float(new_d_noise)
# --------
# measurement_prob
# computes the probability of a measurement
#
def measurement_prob(self, measurements):
# calculate the correct measurement
predicted_measurements = self.sense(0) # Our sense function took 0 as an argument to switch off noise.
# compute errors
error = 1.0
for i in range(len(measurements)):
error_bearing = abs(measurements[i] - predicted_measurements[i])
error_bearing = (error_bearing + pi) % (2.0 * pi) - pi # truncate
# update Gaussian
error *= (exp(- (error_bearing ** 2) / (self.bearing_noise ** 2) / 2.0) /
sqrt(2.0 * pi * (self.bearing_noise ** 2)))
return error
def __repr__(self): #allows us to print robot attributes.
return '[x=%.6s y=%.6s orient=%.6s]' % (str(self.x), str(self.y),
str(self.orientation))
############# ONLY ADD/MODIFY CODE BELOW HERE ###################
# --------
# move:
#
def move(self, motion): # Do not change the name of this function
steering=motion[0]
distance=motion[1]
if abs(steering)>max_steering_angle:
raise ValueError, 'Exceeding max steering angle'
if distance<0.0:
raise ValueError, 'Moving backwards is not valid'
result= robot(self.length)
result.bearing_noise=self.bearing_noise
result.steering_noise=self.steering_noise
result.distance_noise=self.distance_noise
steering2=random.gauss(steering,self.steering_noise)
distance2=random.gauss(distance,self.distance_noise)
turn=distance2*tan(steering2)/result.length
if abs(turn)<0.01:
result.orientation=(self.orientation+turn)%(2.0*pi)
result.x=self.x+distance2*cos(self.orientation)
result.y=self.y+distance2*sin(self.orientation)
else:
R=result.length/tan(steering2) #或者=distance2/turn
cx=self.x-sin(self.orientation)*R
cy=self.y+cos(self.orientation)*R
result.orientation=(self.orientation+turn)%(2.0*pi)
result.x=cx+sin(result.orientation)*R
result.y=cy-cos(result.orientation)*R
return result
# --------
# sense:
#
def sense(self,add_noise=1): #do not change the name of this function
Z = []
for i in range(len(landmarks)):
bearing=atan2(landmarks[i][0]-self.y,landmarks[i][1]-self.x)-self.orientation
if add_noise:
bearing+=random.gauss(0.0,self.bearing_noise)
bearing%=2*pi
Z.append(bearing)
return Z
############## ONLY ADD/MODIFY CODE ABOVE HERE ####################
# --------
#
# extract position from a particle set
#
def get_position(p):
x = 0.0
y = 0.0
orientation = 0.0
for i in range(len(p)):
x += p[i].x
y += p[i].y
# orientation is tricky because it is cyclic. By normalizing
# around the first particle we are somewhat more robust to
# the 0=2pi problem
orientation += (((p[i].orientation - p[0].orientation + pi) % (2.0 * pi))
+ p[0].orientation - pi)
return [x / len(p), y / len(p), orientation / len(p)]
# --------
#
# The following code generates the measurements vector
# You can use it to develop your solution.
#
def generate_ground_truth(motions):
myrobot = robot()
myrobot.set_noise(bearing_noise, steering_noise, distance_noise)
Z = []
T = len(motions)
for t in range(T):
myrobot = myrobot.move(motions[t])
Z.append(myrobot.sense())
#print 'Robot: ', myrobot
return [myrobot, Z]
# --------
#
# The following code prints the measurements associated
# with generate_ground_truth
#
def print_measurements(Z):
T = len(Z)
print 'measurements = [[%.8s, %.8s, %.8s, %.8s],' % \
(str(Z[0][0]), str(Z[0][1]), str(Z[0][2]), str(Z[0][3]))
for t in range(1,T-1):
print ' [%.8s, %.8s, %.8s, %.8s],' % \
(str(Z[t][0]), str(Z[t][1]), str(Z[t][2]), str(Z[t][3]))
print ' [%.8s, %.8s, %.8s, %.8s]]' % \
(str(Z[T-1][0]), str(Z[T-1][1]), str(Z[T-1][2]), str(Z[T-1][3]))
# --------
#
# The following code checks to see if your particle filter
# localizes the robot to within the desired tolerances
# of the true position. The tolerances are defined at the top.
#
def check_output(final_robot, estimated_position):
error_x = abs(final_robot.x - estimated_position[0])
error_y = abs(final_robot.y - estimated_position[1])
error_orientation = abs(final_robot.orientation - estimated_position[2])
error_orientation = (error_orientation + pi) % (2.0 * pi) - pi
correct = error_x < tolerance_xy and error_y < tolerance_xy \
and error_orientation < tolerance_orientation
return correct
def particle_filter(motions, measurements, N=500): # I know it's tempting, but don't change N!
# --------
#
# Make particles
#
p = []
for i in range(N):
r = robot()
r.set_noise(bearing_noise, steering_noise, distance_noise)
p.append(r)
# --------
#
# Update particles
#
for t in range(len(motions)):
# motion update (prediction)
p2 = []
for i in range(N):
p2.append(p[i].move(motions[t]))
p = p2
# measurement update
w = []
for i in range(N):
w.append(p[i].measurement_prob(measurements[t]))
# resampling
p3 = []
index = int(random.random() * N)
beta = 0.0
mw = max(w)
for i in range(N):
beta += random.random() * 2.0 * mw
while beta > w[index]:
beta -= w[index]
index = (index + 1) % N
p3.append(p[index])
p = p3
return get_position(p)
## --------
## TEST CASES:
##
##1) Calling the particle_filter function with the following
## motions and measurements should return a [x,y,orientation]
## vector near [x=93.476 y=75.186 orient=5.2664], that is, the
## robot's true location.
##
motions = [[2. * pi / 10, 20.] for row in range(8)]
measurements = [[4.746936, 3.859782, 3.045217, 2.045506],
[3.510067, 2.916300, 2.146394, 1.598332],
[2.972469, 2.407489, 1.588474, 1.611094],
[1.906178, 1.193329, 0.619356, 0.807930],
[1.352825, 0.662233, 0.144927, 0.799090],
[0.856150, 0.214590, 5.651497, 1.062401],
[0.194460, 5.660382, 4.761072, 2.471682],
[5.717342, 4.736780, 3.909599, 2.342536]]
print particle_filter(motions, measurements)
## 2) You can generate your own test cases by generating
## measurements using the generate_ground_truth function.
## It will print the robot's last location when calling it.
##
##
##number_of_iterations = 6
##motions = [[2. * pi / 20, 12.] for row in range(number_of_iterations)]
##
##x = generate_ground_truth(motions)
##final_robot = x[0]
##measurements = x[1]
##estimated_position = particle_filter(motions, measurements)
##print_measurements(measurements)
##print 'Ground truth: ', final_robot
##print 'Particle filter: ', estimated_position
##print 'Code check: ', check_output(final_robot, estimated_position)