Udacity机器人软件工程师课程笔记（三十三)

一、概述

之前的文章介绍过卡尔曼滤波算法进行定位，我们知道kalman算法适合用于线性的高斯分布的状态环境中，我们也介绍了EKF，来解决在非高斯和非线性环境下的机器人定位算法。但是他们在现实应用中存在计算量，内存消耗上不是很高效。这就引出了MCL算法。

粒子滤波很粗浅的说就是一开始在地图空间很均匀的撒一把粒子，然后通过获取机器人的motion来移动粒子，比如机器人向前移动了一米，所有的粒子也就向前移动一米，不管现在这个粒子的位置对不对。使用每个粒子所处位置模拟一个传感器信息跟观察到的传感器信息（一般是激光）作对比，从而赋给每个粒子一个概率。之后根据生成的概率来重新生成粒子，概率越高的生成的概率越大。这样的迭代之后，所有的粒子会慢慢地收敛到一起，机器人的确切位置也就被推算出来了

MCL的计算步骤：

随机产生一堆粒子：粒子可以有位置、方向和/或任何其他需要估计的状态变量。每一个都有一个权值(概率)，表明它与系统的实际状态匹配的可能性有多大。用相同的权重初始化每个变量。
预测粒子的下一个状态：根据真实系统行为的预测来移动粒子。
更新：根据测量结果更新粒子的权重。与测量值密切匹配的粒子的权重要高于与测量值不太匹配的粒子。
重新采样：抛弃高度不可能的粒子，代之以更可能的粒子的副本。
计算估计值：可选地，计算粒子集的加权平均值和协方差来得到状态估计。

粒子滤波的基本步骤为上面所述的5步，其本质是使用一组有限的加权随机样本（粒子）来近似表征任意状态的后验概率密度。粒子滤波的优势在于对复杂问题的求解上，比如高度的非线性、非高斯动态系统的状态递推估计或概率推理问题。

此部分转自知乎专栏

二、MCL算法

蒙特卡罗定位算法的伪代码如上图所示，其由两个主要部分组成由两个for循环表示。第一个部分是运动和传感器更新，第二个是重采样过程。

若给出一张环境地图，MCL的目标是确定由可信度代表的机器人姿态。在每次迭代中算法都采用之前的可信度作为启动命令，传感器测量作为输入。

最初，可信度是通过随机生成m个粒子获得的。然后，在第一个for循环中，假设的状态是在机器人移动时计算出来的。接下来，用新的传感器测量值计算粒子的权重，然后更新每一个粒子的状态。

在MCL的第二个循环中进行重采样，在这里具有高概率的粒子存活下来并在下一次迭代中被重新绘制出来，低概率粒子则被抛弃。

最后算法输出新的可信度，然后启动新的迭代，通过读取新的传感器测量值来实现下一个运动。

三、C++代码实现

以下是基础代码，我们需要在mian中完成相关程序

#define _USE_MATH_DEFINES
//#include "src/matplotlibcpp.h"//Graph Library
#include <iostream>
#include <string>
#include <math.h>
#include <vector>
#include <stdexcept> // throw errors
#include <random> //C++ 11 Random Numbers//namespace plt = matplotlibcpp;
using namespace std;// Landmarks
double landmarks[8][2] = { { 20.0, 20.0 }, { 20.0, 80.0 }, { 20.0, 50.0 },{ 50.0, 20.0 }, { 50.0, 80.0 }, { 80.0, 80.0 },{ 80.0, 20.0 }, { 80.0, 50.0 } };// Map size in meters
double world_size = 100.0;// Random Generators
random_device rd;
mt19937 gen(rd());// Global Functions
double mod(double first_term, double second_term);
double gen_real_random();class Robot {public:Robot(){// Constructorx = gen_real_random() * world_size; // robot's x coordinatey = gen_real_random() * world_size; // robot's y coordinateorient = gen_real_random() * 2.0 * M_PI; // robot's orientationforward_noise = 0.0; //noise of the forward movementturn_noise = 0.0; //noise of the turnsense_noise = 0.0; //noise of the sensing}void set(double new_x, double new_y, double new_orient){// Set robot new position and orientationif (new_x < 0 || new_x >= world_size)throw std::invalid_argument("X coordinate out of bound");if (new_y < 0 || new_y >= world_size)throw std::invalid_argument("Y coordinate out of bound");if (new_orient < 0 || new_orient >= 2 * M_PI)throw std::invalid_argument("Orientation must be in [0..2pi]");x = new_x;y = new_y;orient = new_orient;}void set_noise(double new_forward_noise, double new_turn_noise, double new_sense_noise){// Simulate noise, often useful in particle filtersforward_noise = new_forward_noise;turn_noise = new_turn_noise;sense_noise = new_sense_noise;}vector<double> sense(){// Measure the distances from the robot toward the landmarksvector<double> z(sizeof(landmarks) / sizeof(landmarks[0]));double dist;for (int i = 0; i < sizeof(landmarks) / sizeof(landmarks[0]); i++) {dist = sqrt(pow((x - landmarks[i][0]), 2) + pow((y - landmarks[i][1]), 2));dist += gen_gauss_random(0.0, sense_noise);z[i] = dist;}return z;}Robot move(double turn, double forward){if (forward < 0)throw std::invalid_argument("Robot cannot move backward");// turn, and add randomness to the turning commandorient = orient + turn + gen_gauss_random(0.0, turn_noise);orient = mod(orient, 2 * M_PI);// move, and add randomness to the motion commanddouble dist = forward + gen_gauss_random(0.0, forward_noise);x = x + (cos(orient) * dist);y = y + (sin(orient) * dist);// cyclic truncatex = mod(x, world_size);y = mod(y, world_size);// set particleRobot res;res.set(x, y, orient);res.set_noise(forward_noise, turn_noise, sense_noise);return res;}string show_pose(){// Returns the robot current position and orientation in a string formatreturn "[x=" + to_string(x) + " y=" + to_string(y) + " orient=" + to_string(orient) + "]";}string read_sensors(){// Returns all the distances from the robot toward the landmarksvector<double> z = sense();string readings = "[";for (int i = 0; i < z.size(); i++) {readings += to_string(z[i]) + " ";}readings[readings.size() - 1] = ']';return readings;}double measurement_prob(vector<double> measurement){// Calculates how likely a measurement should bedouble prob = 1.0;double dist;for (int i = 0; i < sizeof(landmarks) / sizeof(landmarks[0]); i++) {dist = sqrt(pow((x - landmarks[i][0]), 2) + pow((y - landmarks[i][1]), 2));prob *= gaussian(dist, sense_noise, measurement[i]);}return prob;}double x, y, orient; //robot posesdouble forward_noise, turn_noise, sense_noise; //robot noisesprivate:double gen_gauss_random(double mean, double variance){// Gaussian randomnormal_distribution<double> gauss_dist(mean, variance);return gauss_dist(gen);}double gaussian(double mu, double sigma, double x){// Probability of x for 1-dim Gaussian with mean mu and var. sigmareturn exp(-(pow((mu - x), 2)) / (pow(sigma, 2)) / 2.0) / sqrt(2.0 * M_PI * (pow(sigma, 2)));}
};// Functions
double gen_real_random()
{// Generate real random between 0 and 1uniform_real_distribution<double> real_dist(0.0, 1.0); //Realreturn real_dist(gen);
}double mod(double first_term, double second_term)
{// Compute the modulusreturn first_term - (second_term)*floor(first_term / (second_term));
}double evaluation(Robot r, Robot p[], int n)
{//Calculate the mean error of the systemdouble sum = 0.0;for (int i = 0; i < n; i++) {//the second part is because of world's cyclicitydouble dx = mod((p[i].x - r.x + (world_size / 2.0)), world_size) - (world_size / 2.0);double dy = mod((p[i].y - r.y + (world_size / 2.0)), world_size) - (world_size / 2.0);double err = sqrt(pow(dx, 2) + pow(dy, 2));sum += err;}return sum / n;
}
double max(double arr[], int n)
{// Identify the max element in an arraydouble max = 0;for (int i = 0; i < n; i++) {if (arr[i] > max)max = arr[i];}return max;
}int main()
{...
}

（1）运动和传感器更新

首先完成伪代码的第一个部分：

主函数如下，

int main()
{//Practice Interfacing with Robot ClassRobot myrobot;myrobot.set_noise(5.0, 0.1, 5.0);myrobot.set(30.0, 50.0, M_PI / 2.0);myrobot.move(-M_PI / 2.0, 15.0);//cout << myrobot.read_sensors() << endl;myrobot.move(-M_PI / 2.0, 10.0);//cout << myrobot.read_sensors() << endl;// 实例化1000个粒子，每个粒子具有随机的位置和方向int n = 1000;Robot p[1000];//For each particle add noise: Forward_Noise=0.05, Turn_Noise=0.05, and Sense_Noise=5.0for (int i = 0; i < n; i++) {p[i].set_noise(0.05, 0.05, 5.0);cout << p[i].show_pose() << endl;}//重新初始化myrobot对象并初始化一个测量向量myrobot = Robot();vector<double> z;//移动机器人，然后感知环境myrobot = myrobot.move(0.1, 5.0);z = myrobot.sense();// 以机器人运动模拟每个粒子的运动Robot p2[1000];for (int i = 0; i < n; i++) {p2[i] = p[i].move(0.1, 5.0);p[i] = p2[i];}//根据机器人的测量产生粒子的权重double w[1000];for (int i = 0; i < n; i++) {w[i] = p[i].measurement_prob(z);cout << w[i] << endl;}return 0;
}

（2）重采样

来看一组只有五个数值的例子，

来计算归一化的各个粒子的概率

#include <iostream>using namespace std;double w[] = { 0.6, 1.2, 2.4, 0.6, 1.2 };
double sum = 0;void ComputeProb(double w[], int n)
{// 计算总权重Wfor (int i = 0; i < n; i++) {sum = sum + w[i];}// 计算归一化后的权重for (int j = 0; j < n; j++) {w[j] = w[j] / sum;cout << "P" << j + 1 << "=" << w[j] << endl;}
}int main()
{ComputeProb(w, sizeof(w) / sizeof(w[0]));return 0;
}

重采样伪代码如下

多次重采样数据，再利用评估函数来评估误差

int main()
{//Practice Interfacing with Robot ClassRobot myrobot;myrobot.set_noise(5.0, 0.1, 5.0);myrobot.set(30.0, 50.0, M_PI / 2.0);myrobot.move(-M_PI / 2.0, 15.0);myrobot.move(-M_PI / 2.0, 10.0);// 实例化1000个粒子，每个粒子具有随机的位置和方向int n = 1000;Robot p[1000];//For each particle add noise: Forward_Noise=0.05, Turn_Noise=0.05, and Sense_Noise=5.0for (int i = 0; i < n; i++) {p[i].set_noise(0.05, 0.05, 5.0);// cout << p[i].show_pose() << endl;}// 重新初始化myrobot对象并初始化一个测量向量myrobot = Robot();vector<double> z;// 在一组粒子上迭代50次int steps = 50;for (int t = 0; t < steps; t++) {// 移动机器人，然后感知环境myrobot = myrobot.move(0.1, 5.0);z = myrobot.sense();// 模拟每个粒子的机器人运动Robot p2[1000];for (int i = 0; i < n; i++) {p2[i] = p[i].move(0.1, 5.0);p[i] = p2[i];}// 根据机器人的测量产生粒子的权重double w[1000];for (int i = 0; i < n; i++) {w[i] = p[i].measurement_prob(z);//cout << w[i] << endl;}// 对粒子重新采样，采样概率与重要性权重成正比Robot p3[1000];int index = gen_real_random() * n;//cout << index << endl;double beta = 0.0;double mw = max(w, n);//cout << mw;for (int i = 0; i < n; i++) {beta += gen_real_random() * 2.0 * mw;while (beta > w[index]) {beta -= w[index];index = mod((index + 1), n);}p3[i] = p[index];}for (int k = 0; k < n; k++) {p[k] = p3[k];//cout << p[k].show_pose() << endl;}// 评估误差cout << "Step = " << t << ", Evaluation = " << evaluation(myrobot, p, n) << endl;} //End of Steps loopreturn 0;
}

输出如下

Step = 0, Evaluation = 3.22026
Step = 1, Evaluation = 3.3267
Step = 2, Evaluation = 3.6514
Step = 3, Evaluation = 4.42686
Step = 4, Evaluation = 3.97611
Step = 5, Evaluation = 3.1907
Step = 6, Evaluation = 2.56729
Step = 7, Evaluation = 2.15039
Step = 8, Evaluation = 1.83402
Step = 9, Evaluation = 1.48423
Step = 10, Evaluation = 1.4108
Step = 11, Evaluation = 1.39957
Step = 12, Evaluation = 1.40748
Step = 13, Evaluation = 1.37557
Step = 14, Evaluation = 1.33284
Step = 15, Evaluation = 1.36003
Step = 16, Evaluation = 1.44411
Step = 17, Evaluation = 1.55345
Step = 18, Evaluation = 1.60188
Step = 19, Evaluation = 1.53727
Step = 20, Evaluation = 1.48127
Step = 21, Evaluation = 1.43857
Step = 22, Evaluation = 1.37033
Step = 23, Evaluation = 1.3693
Step = 24, Evaluation = 1.42263
Step = 25, Evaluation = 1.43937
Step = 26, Evaluation = 1.40299
Step = 27, Evaluation = 1.39867
Step = 28, Evaluation = 1.42217
Step = 29, Evaluation = 1.41403
Step = 30, Evaluation = 1.44112
Step = 31, Evaluation = 1.43527
Step = 32, Evaluation = 1.41816
Step = 33, Evaluation = 1.45077
Step = 34, Evaluation = 1.51069
Step = 35, Evaluation = 1.58236
Step = 36, Evaluation = 1.47355
Step = 37, Evaluation = 1.40463
Step = 38, Evaluation = 1.41416
Step = 39, Evaluation = 1.40608
Step = 40, Evaluation = 1.44435
Step = 41, Evaluation = 1.47949
Step = 42, Evaluation = 1.53257
Step = 43, Evaluation = 1.56387
Step = 44, Evaluation = 1.52004
Step = 45, Evaluation = 1.45646
Step = 46, Evaluation = 1.42782
Step = 47, Evaluation = 1.439
Step = 48, Evaluation = 1.42743
Step = 49, Evaluation = 1.40226

四、绘图

以下内容在虚拟机Linux环境中完成。

下载udacity提供的源文件。

修改main.cpp。

//Compile with: g++ solution.cpp -o app -std=c++11 -I/usr/include/python2.7 -lpython2.7
#include "src/matplotlibcpp.h" //Graph Library
#include <iostream>
#include <string>
#include <math.h>
#include <stdexcept> // throw errors
#include <random> //C++ 11 Random Numbers
#include <vector>namespace plt = matplotlibcpp;
using namespace std;// Landmarks
double landmarks[8][2] = { { 20.0, 20.0 }, { 20.0, 80.0 }, { 20.0, 50.0 },{ 50.0, 20.0 }, { 50.0, 80.0 }, { 80.0, 80.0 },{ 80.0, 20.0 }, { 80.0, 50.0 } };// Map size in meters
double world_size = 100.0;// Random Generators
random_device rd;
mt19937 gen(rd());// Global Functions
double mod(double first_term, double second_term);
double gen_real_random();class Robot {public:Robot(){// Constructorx = gen_real_random() * world_size; // robot's x coordinatey = gen_real_random() * world_size; // robot's y coordinateorient = gen_real_random() * 2.0 * M_PI; // robot's orientationforward_noise = 0.0; //noise of the forward movementturn_noise = 0.0; //noise of the turnsense_noise = 0.0; //noise of the sensing}void set(double new_x, double new_y, double new_orient){// Set robot new position and orientationif (new_x < 0 || new_x >= world_size)throw std::invalid_argument("X coordinate out of bound");if (new_y < 0 || new_y >= world_size)throw std::invalid_argument("Y coordinate out of bound");if (new_orient < 0 || new_orient >= 2 * M_PI)throw std::invalid_argument("Orientation must be in [0..2pi]");x = new_x;y = new_y;orient = new_orient;}void set_noise(double new_forward_noise, double new_turn_noise, double new_sense_noise){// Simulate noise, often useful in particle filtersforward_noise = new_forward_noise;turn_noise = new_turn_noise;sense_noise = new_sense_noise;}vector<double> sense(){// Measure the distances from the robot toward the landmarksvector<double> z(sizeof(landmarks) / sizeof(landmarks[0]));double dist;for (int i = 0; i < sizeof(landmarks) / sizeof(landmarks[0]); i++) {dist = sqrt(pow((x - landmarks[i][0]), 2) + pow((y - landmarks[i][1]), 2));dist += gen_gauss_random(0.0, sense_noise);z[i] = dist;}return z;}Robot move(double turn, double forward){if (forward < 0)throw std::invalid_argument("Robot cannot move backward");// turn, and add randomness to the turning commandorient = orient + turn + gen_gauss_random(0.0, turn_noise);orient = mod(orient, 2 * M_PI);// move, and add randomness to the motion commanddouble dist = forward + gen_gauss_random(0.0, forward_noise);x = x + (cos(orient) * dist);y = y + (sin(orient) * dist);// cyclic truncatex = mod(x, world_size);y = mod(y, world_size);// set particleRobot res;res.set(x, y, orient);res.set_noise(forward_noise, turn_noise, sense_noise);return res;}string show_pose(){// Returns the robot current position and orientation in a string formatreturn "[x=" + to_string(x) + " y=" + to_string(y) + " orient=" + to_string(orient) + "]";}string read_sensors(){// Returns all the distances from the robot toward the landmarksvector<double> z = sense();string readings = "[";for (int i = 0; i < z.size(); i++) {readings += to_string(z[i]) + " ";}readings[readings.size() - 1] = ']';return readings;}double measurement_prob(vector<double> measurement){// Calculates how likely a measurement should bedouble prob = 1.0;double dist;for (int i = 0; i < sizeof(landmarks) / sizeof(landmarks[0]); i++) {dist = sqrt(pow((x - landmarks[i][0]), 2) + pow((y - landmarks[i][1]), 2));prob *= gaussian(dist, sense_noise, measurement[i]);}return prob;}double x, y, orient; //robot posesdouble forward_noise, turn_noise, sense_noise; //robot noisesprivate:double gen_gauss_random(double mean, double variance){// Gaussian randomnormal_distribution<double> gauss_dist(mean, variance);return gauss_dist(gen);}double gaussian(double mu, double sigma, double x){// Probability of x for 1-dim Gaussian with mean mu and var. sigmareturn exp(-(pow((mu - x), 2)) / (pow(sigma, 2)) / 2.0) / sqrt(2.0 * M_PI * (pow(sigma, 2)));}
};// Functions
double gen_real_random()
{// Generate real random between 0 and 1uniform_real_distribution<double> real_dist(0.0, 1.0); //Realreturn real_dist(gen);
}double mod(double first_term, double second_term)
{// Compute the modulusreturn first_term - (second_term)*floor(first_term / (second_term));
}double evaluation(Robot r, Robot p[], int n)
{//Calculate the mean error of the systemdouble sum = 0.0;for (int i = 0; i < n; i++) {//the second part is because of world's cyclicitydouble dx = mod((p[i].x - r.x + (world_size / 2.0)), world_size) - (world_size / 2.0);double dy = mod((p[i].y - r.y + (world_size / 2.0)), world_size) - (world_size / 2.0);double err = sqrt(pow(dx, 2) + pow(dy, 2));sum += err;}return sum / n;
}
double max(double arr[], int n)
{// Identify the max element in an arraydouble max = 0;for (int i = 0; i < n; i++) {if (arr[i] > max)max = arr[i];}return max;
}void visualization(int n, Robot robot, int step, Robot p[], Robot pr[])
{//Draw the robot, landmarks, particles and resampled particles on a graph//Graph Formatplt::title("MCL, step " + to_string(step));plt::xlim(0, 100);plt::ylim(0, 100);//Draw particles in greenfor (int i = 0; i < n; i++) {plt::plot({ p[i].x }, { p[i].y }, "go");}//Draw resampled particles in yellowfor (int i = 0; i < n; i++) {plt::plot({ pr[i].x }, { pr[i].y }, "yo");}//Draw landmarks in redfor (int i = 0; i < sizeof(landmarks) / sizeof(landmarks[0]); i++) {plt::plot({ landmarks[i][0] }, { landmarks[i][1] }, "ro");}//Draw robot position in blueplt::plot({ robot.x }, { robot.y }, "bo");//Save the image and close the plotplt::save("./Images/Step" + to_string(step) + ".png");plt::clf();
}int main()
{//Practice Interfacing with Robot ClassRobot myrobot;myrobot.set_noise(5.0, 0.1, 5.0);myrobot.set(30.0, 50.0, M_PI / 2.0);myrobot.move(-M_PI / 2.0, 15.0);//cout << myrobot.read_sensors() << endl;myrobot.move(-M_PI / 2.0, 10.0);//cout << myrobot.read_sensors() << endl;// Create a set of particlesint n = 1000;Robot p[n];for (int i = 0; i < n; i++) {p[i].set_noise(0.05, 0.05, 5.0);//cout << p[i].show_pose() << endl;}//Re-initialize myrobot object and Initialize a measurment vectormyrobot = Robot();vector<double> z;//Iterating 50 times over the set of particlesint steps = 50;for (int t = 0; t < steps; t++) {//Move the robot and sense the environment afterwardsmyrobot = myrobot.move(0.1, 5.0);z = myrobot.sense();// Simulate a robot motion for each of these particlesRobot p2[n];for (int i = 0; i < n; i++) {p2[i] = p[i].move(0.1, 5.0);p[i] = p2[i];}//Generate particle weights depending on robot's measurementdouble w[n];for (int i = 0; i < n; i++) {w[i] = p[i].measurement_prob(z);//cout << w[i] << endl;}//Resample the particles with a sample probability proportional to the importance weightRobot p3[n];int index = gen_real_random() * n;//cout << index << endl;double beta = 0.0;double mw = max(w, n);//cout << mw;for (int i = 0; i < n; i++) {beta += gen_real_random() * 2.0 * mw;while (beta > w[index]) {beta -= w[index];index = mod((index + 1), n);}p3[i] = p[index];}for (int k = 0; k < n; k++) {p[k] = p3[k];//cout << p[k].show_pose() << endl;}//Evaluate the Errorcout << "Step = " << t << ", Evaluation = " << evaluation(myrobot, p, n) << endl;//####   DON'T MODIFY ANYTHING ABOVE HERE! ENTER CODE BELOW ####//TODO: Graph the position of the robot and the particles at each stepif (t % 5 == 0 ){visualization(n, myrobot, t, p2, p);}} //End of Steps loopreturn 0;
}

编译程序

$ g ++ main.cpp -o app -std = c ++ 11 -I / usr / include / python2.7 -lpython2.7

运行程序

./app

效果如图