粒子濾波（Particle Filter）及其在機器人定位中的應用

一、粒子濾波簡介

粒子濾波的基本思想是用随機樣本來描述機率分布，然後在測量的基礎上，通過調節各粒子權值的大小和樣本的位置，來近似實際機率分布，并以樣本的均值作為系統的估計值。  

粒子濾波是指：通過尋找一組在狀态空間中傳播的随機樣本對機率密度函數P(Xk|Zk)進行近似，以樣本均值代替積分運算，進而獲得狀态最小方差估計的過程，這些樣本即稱為“粒子。采用數學語言描述如下：對于平穩的随機過程，假定k-1時刻系統的後驗機率密度為P(Xk-1|Zk-1)，依據一定原則選取n個随機樣本點，k時刻獲得測量資訊後，經過狀态和時間更新過程，n個粒子的後驗機率密度可近似為P(Xk|Zk) 。随着粒子數目的增加，粒子的機率密度函數逐漸逼近狀态的機率密度數，粒子濾波估計即達到了最優貝葉斯估計的效果。

粒子濾波的算法步驟如下：

粒子濾波的推導過程詳見http://blog.csdn.net/heyijia0327/article/category/3245449

二、粒子濾波在機器人定位中的應用

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)