其中，表示节点集合，表示边集。详细说明过程已在上一小节给出。

1.4 RRT算法的优缺点

优点

RRT算法适用范围广、参数简单、高维空间规划性能优秀。既能够用于机械臂的运动力学规划，也可用于机器人或无人机等进行路径规划。

在使用 RRT算法进行路径规划时，若能够获得全局环境并进行建模，可进行全局路径规划。若无法获得全局环境，如自动驾驶汽车路径规划问题，能够在动态规划中对局部地图进行规划以生成局部路径，也为无人机等高维空间的路径规划提供了可行方案。

缺点

在扩展节点时从无障碍区域内随机选择节点，会导致产生部分无用节点，节点利用率低，增加算法随机性的同时也降低了算法的收敛速度。

由于随机树扩展时会判断 和 连线方向上有无障碍物，若有障碍物则会放弃在该方向上扩展节点。 因此当路径中包含障碍物之间形成的狭窄通道时，使用RRT算法规划路径有一定几率无法规划出最优路径。

2. python代码实现

代码主体部分来自pythonRobotics，这边只进行了相应的注释和部分代码的修改。

import math
import random

import matplotlib.pyplot as plt
import numpy as np
from celluloid import Camera  # 保存动图时用，pip install celluloid




class RRT:
    """
    Class for RRT planning
    """

    class Node:
        """
        创建节点
        """

        def __init__(self, x, y):
            self.x = x  # 节点坐标
            self.y = y
            self.path_x = [] # 路径，作为画图的数据
            self.path_y = []
            self.parent = None #父节点

    class AreaBounds:
        """区域大小
        """
        def __init__(self, area):
            self.xmin = float(area[0])
            self.xmax = float(area[1])
            self.ymin = float(area[2])
            self.ymax = float(area[3])

    def __init__(self,
            start,
            goal,
            obstacle_list,
            rand_area,
            expand_dis=3.0,
            goal_sample_rate=5,
            max_iter=500,
            play_area=None,
            robot_radius=0.0,
            ):
        """
        Setting Parameter

        start:起点 [x,y]
        goal:目标点 [x,y]
        obstacleList:障碍物位置列表 [[x,y,size],...]
        rand_area: 采样区域 x,y ∈ [min,max]
        play_area: 约束随机树的范围 [xmin,xmax,ymin,ymax]
        robot_radius: 机器人半径
        expand_dis: 扩展的步长
        goal_sample_rate: 采样目标点的概率，百分制.default: 5，即表示5%的概率直接采样目标点

        """
        self.start = self.Node(start[0], start[1]) # 根节点
        self.end = self.Node(goal[0], goal[1]) 
        self.min_rand = rand_area[0]
        self.max_rand = rand_area[1]
        if play_area is not None:
            self.play_area = self.AreaBounds(play_area)
        else:
            self.play_area = None
        self.expand_dis = expand_dis
        self.goal_sample_rate = goal_sample_rate
        self.max_iter = max_iter
        self.obstacle_list = obstacle_list
        self.node_list = []
        self.robot_radius = robot_radius

    def planning(self, animation=True,camara=None):
        """
        rrt path planning

        animation: flag for animation on or off

        camara: 是否保存动图
        """

        # 将起点作为根节点x_{init}​，加入到随机树的节点集合中。
        self.node_list = [self.start]
        for i in range(self.max_iter):
            # 从可行区域内随机选取一个节点x_{rand}
            rnd_node = self.sample_free()  

            # 已生成的树中利用欧氏距离判断距离x_{rand}​最近的点x_{near}。
            nearest_ind = self.get_nearest_node_index(self.node_list, rnd_node)
            nearest_node = self.node_list[nearest_ind]  

            # 从x_{near}与x_{rand}的连线方向上扩展固定步长u，得到新节点 x_{new}
            new_node = self.steer(nearest_node, rnd_node, self.expand_dis)
            
            # 如果在可行区域内，且x_{near}与x_{new}之间无障碍物
            if self.is_inside_play_area(new_node, self.play_area) and \
               self.obstacle_free(new_node, self.obstacle_list, self.robot_radius):
                self.node_list.append(new_node)


            
            # 如果此时得到的节点x_new到目标点的距离小于扩展步长，则直接将目标点作为x_rand。
            if self.calc_dist_to_goal(self.node_list[-1].x,self.node_list[-1].y) <= self.expand_dis:
                final_node = self.steer(self.node_list[-1], self.end,self.expand_dis)
                if self.obstacle_free(final_node, self.obstacle_list, self.robot_radius):
                    # 返回最终路径
                    return self.generate_final_course(len(self.node_list) - 1)

            if animation and i % 5 ==0:
                self.draw_graph(rnd_node, camara)

        return None  # cannot find path

    def steer(self, from_node, to_node, extend_length=float("inf")):
        """连线方向扩展固定步长查找x_new

        Args:
            from_node (_type_): x_near
            to_node (_type_): x_rand
            extend_length (_type_, optional): 扩展步长u. Defaults to float("inf").

        Returns:
            _type_: _description_
        """
        # 利用反正切计算角度, 然后利用角度和步长计算新坐标
        d, theta = self.calc_distance_and_angle(from_node, to_node)

        # 如果$x_{near}$与$x_{rand}$间的距离小于步长，则直接将$x_{rand}$作为新节点$x_{new}$
        if extend_length >= d:
            new_x = to_node.x
            new_y = to_node.y
        else:
            new_x = from_node.x+math.cos(theta)
            new_y = from_node.y+math.sin(theta)
        new_node = self.Node(new_x,new_y)
        new_node.path_x = [from_node.x]
        new_node.path_y = [from_node.y]
        new_node.path_x.append(new_x)
        new_node.path_y.append(new_y)

        new_node.parent = from_node

        return new_node



    def generate_final_course(self, goal_ind):
        """生成路径
        Args:
            goal_ind (_type_): 目标点索引

        Returns:
            _type_: _description_
        """
        path = [[self.end.x, self.end.y]]
        node = self.node_list[goal_ind]
        while node.parent is not None:
            path.append([node.x, node.y])
            node = node.parent
        path.append([node.x, node.y])

        return path

    def calc_dist_to_goal(self, x, y):
        """计算(x,y)离目标点的距离
        """
        dx = x - self.end.x
        dy = y - self.end.y
        return math.hypot(dx, dy)

    def sample_free(self):
        # 以（100-goal_sample_rate）%的概率随机生长，(goal_sample_rate)%的概率朝向目标点生长
        if random.randint(0, 100) > self.goal_sample_rate:
            rnd = self.Node(
                random.uniform(self.min_rand, self.max_rand),
                random.uniform(self.min_rand, self.max_rand))
        else:  # goal point sampling
            rnd = self.Node(self.end.x, self.end.y)
        return rnd

    def draw_graph(self, rnd=None, camera=None):
        if camera==None:
            plt.clf()
        # 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])
        # 画随机点
        if rnd is not None:
            plt.plot(rnd.x, rnd.y, "^k")
            if self.robot_radius > 0.0:
                self.plot_circle(rnd.x, rnd.y, self.robot_radius, '-r')
        # 画已生成的树
        for node in self.node_list:
            if node.parent:
                plt.plot(node.path_x, node.path_y, "-g")

        # 画障碍物
        for (ox, oy, size) in self.obstacle_list:
            self.plot_circle(ox, oy, size)

        # 如果约定了可行区域，则画出可行区域
        if self.play_area is not None:
            plt.plot([self.play_area.xmin, self.play_area.xmax,
                self.play_area.xmax, self.play_area.xmin,
                self.play_area.xmin],
                [self.play_area.ymin, self.play_area.ymin,
                self.play_area.ymax, self.play_area.ymax,
                self.play_area.ymin],
                "-k")

        # 画出起点和目标点
        plt.plot(self.start.x, self.start.y, "xr")
        plt.plot(self.end.x, self.end.y, "xr")
        plt.axis("equal")
        plt.axis([-2, 15, -2, 15])
        plt.grid(True)
        plt.pause(0.01)
        if camera!=None:
            camera.snap()
    # 静态方法无需实例化，也可以实例化后调用，静态方法内部不能调用self.的变量
    @staticmethod
    def plot_circle(x, y, size, color="-b"):  # pragma: no cover
        deg = list(range(0, 360, 5))
        deg.append(0)
        xl = [x + size * math.cos(np.deg2rad(d)) for d in deg]
        yl = [y + size * math.sin(np.deg2rad(d)) for d in deg]
        plt.plot(xl, yl, color)

    @staticmethod
    def get_nearest_node_index(node_list, rnd_node):
        dlist = [(node.x - rnd_node.x)**2 + (node.y - rnd_node.y)**2
                 for node in node_list]
        minind = dlist.index(min(dlist))

        return minind

    @staticmethod
    def is_inside_play_area(node, play_area):

        if play_area is None:
            return True  # no play_area was defined, every pos should be ok

        if node.x < play_area.xmin or node.x > play_area.xmax or \
           node.y < play_area.ymin or node.y > play_area.ymax:
            return False  # outside - bad
        else:
            return True  # inside - ok

    @staticmethod
    def obstacle_free(node, obstacleList, robot_radius):

        if node is None:
            return False

        for (ox, oy, size) in obstacleList:
            dx_list = [ox - x for x in node.path_x]
            dy_list = [oy - y for y in node.path_y]
            d_list = [dx * dx + dy * dy for (dx, dy) in zip(dx_list, dy_list)]

            if min(d_list) <= (size+robot_radius)**2:
                return False  # collision

        return True  # safe

    @staticmethod
    def calc_distance_and_angle(from_node, to_node):
        """计算两个节点间的距离和方位角

        Args:
            from_node (_type_): _description_
            to_node (_type_): _description_

        Returns:
            _type_: _description_
        """
        dx = to_node.x - from_node.x
        dy = to_node.y - from_node.y
        d = math.hypot(dx, dy)
        theta = math.atan2(dy, dx)
        return d, theta


def main(gx=6.0, gy=10.0):
    print("start " + __file__)
    fig = plt.figure(1)

    camera = Camera(fig) # 保存动图时使用
    camera = None # 不保存动图时，camara为None
    show_animation = True
    # ====Search Path with RRT====
    obstacleList = [(5, 5, 1), (3, 6, 2), (3, 8, 2), (3, 10, 2), (7, 5, 2),
                    (9, 5, 2), (8, 10, 1)]  # [x, y, radius]
    # Set Initial parameters
    rrt = RRT(
        start=[0, 0],
        goal=[gx, gy],
        rand_area=[-2, 15],
        obstacle_list=obstacleList,
        play_area=[-2, 12, 0, 14],
        robot_radius=0.8
    )
    path = rrt.planning(animation=show_animation,camara=camera)

    if path is None:
        print("Cannot find path")
    else:
        print("found path!!")

        # Draw final path
        if show_animation:
            rrt.draw_graph(camera=camera)
            plt.grid(True)
            plt.pause(0.01)  
            plt.plot([x for (x, y) in path], [y for (x, y) in path], '-r')
            if camera!=None:
                camera.snap()
                animation = camera.animate()
                animation.save('trajectory.gif')
            plt.show()


if __name__ == '__main__':
    main()

实现效果如下：

所有代码均存于github仓库，欢迎访问。