自动驾驶算法（五）：Informed RRT*算法讲解与代码实现（基于采样的路径规划）与比较

1 RRT*与Informed RRT*

2 Informed RRT*代码解析

3 完整代码

4 算法比较

1 RRT与Informed RRT

        上篇博客我们介绍了RRT*算法：我们在找到一个路径以后我们还会反复的搜索。

        Informed RRT*算法提出的动机(motivation)是能否增加渐近最优的速度呢？

        根据这篇论文的思想：Informed RRT*: Optimal Sampling-based Path Planning Focused via Direct Sampling of an Admissible Ellipsoidal Heuristic。

        我们通过限制采样范围来增加渐进最优的速度。也就是我们在找到路径后限制采样的范围（椭圆），它的长轴就是我们找到路径的cbest，短轴就是如上图，cmin是起点和终点连起来的距离。感兴趣的同学可以去看一下论文。通过cbest越来越小采样的范围也就越来越小。最后我们就会找到路径。我们看一下椭圆是怎么找的，和RRT*相比仅仅改变了采样的Sample函数：

2 Informed RRT*代码解析

        start_time = time.time()self.start = Node(start[0], start[1])self.goal = Node(goal[0], goal[1])self.node_list = [self.start]# max length we expect to find in our 'informed' sample space,# starts as infinitecBest = float('inf')path = None# Computing the sampling spacecMin = math.sqrt(pow(self.start.x - self.goal.x, 2)+ pow(self.start.y - self.goal.y, 2))xCenter = np.array([[(self.start.x + self.goal.x) / 2.0],[(self.start.y + self.goal.y) / 2.0], [0]])a1 = np.array([[(self.goal.x - self.start.x) / cMin],[(self.goal.y - self.start.y) / cMin], [0]])e_theta = math.atan2(a1[1], a1[0])
        这里先初始化cbest为无穷大，将path置为空路径。求出起始点和终点的距离cMin

        Xcenter创建了一个 numpy 数组 xCenter，其中包含了 self.start（导航起始点）和 self.goal（导航终止点） 之间的中心点的 x 和 y 坐标，以及一个零。

        a1 = np.array([[(self.goal.x - self.start.x) / cMin], [(self.goal.y - self.start.y) / cMin], [0]]): 创建了一个 numpy 数组 a1，它表示了从 self.start 指向 self.goal 的单位向量，除以距离 cMin。

        e_theta = math.atan2(a1[1], a1[0]): 使用 math.atan2 计算了单位向量 a1 与 x 轴之间的角度 e_theta（以弧度表示）。

        用数学公式表示如下：
        C = np.array([[math.cos(e_theta), -math.sin(e_theta), 0],[math.sin(e_theta), math.cos(e_theta),  0],[0,                 0,                  1]])
和RRT与RRT*相比，只改变了采样函数，我们看采样函数：
    def informed_sample(self, cMax, cMin, xCenter, C):if cMax < float('inf'):r = [cMax / 2.0,math.sqrt(cMax ** 2 - cMin ** 2) / 2.0,math.sqrt(cMax ** 2 - cMin ** 2) / 2.0]L = np.diag(r)xBall = self.sample_unit_ball()rnd = np.dot(np.dot(C, L), xBall) + xCenterrnd = [rnd[(0, 0)], rnd[(1, 0)]]else:rnd = self.sample()return rnd
    def sample_unit_ball():a = random.random()b = random.random()if b < a:a, b = b, asample = (b * math.cos(2 * math.pi * a / b),b * math.sin(2 * math.pi * a / b))return np.array([[sample[0]], [sample[1]], [0]])
这段代码定义了一个名为 `sample_unit_ball` 的函数，它用于生成一个单位球内的随机点。

具体来说，代码中的操作如下：

1. `a = random.random()`: 生成一个在 [0, 1) 范围内的随机浮点数并将其赋给变量 `a`。

2. `b = random.random()`: 同样地，生成一个在 [0, 1) 范围内的随机浮点数并将其赋给变量 `b`。

3. `if b < a: a, b = b, a`: 如果 `b` 小于 `a`，则交换它们的值，确保 `a` 始终大于等于 `b`。

4. `sample = (b * math.cos(2 * math.pi * a / b), b * math.sin(2 * math.pi * a / b))`: 这里使用生成的随机数 `a` 和 `b` 计算了一个点 `sample`，它位于极坐标系中的角度 `2πa/b` 处，并且距离原点的距离为 `b`。

5. `return np.array([[sample[0]], [sample[1]], [0]])`: 将 `sample` 转换为一个列向量，并添加一个零作为第三个元素，最终返回一个二维的列向量。

总的来说，这个函数用于在单位球内生成一个随机点，它的坐标通过随机生成的参数 `a` 和 `b` 以及一些三角函数计算得到。

rnd = np.dot(np.dot(C, L), xBall) + xCenter这里是先把单位圆压缩成椭圆，得到新采样的点。

3 完整代码

import copy
import math
import random
import timeimport matplotlib.pyplot as plt
from scipy.spatial.transform import Rotation as Rot
import numpy as npshow_animation = Trueclass RRT:# randArea采样范围[-2--18] obstacleList设置的障碍物 maxIter最大迭代次数 expandDis采样步长为2.0 goalSampleRate 以10%的概率将终点作为采样点def __init__(self, obstacleList, randArea,expandDis=2.0, goalSampleRate=10, maxIter=200):self.start = Noneself.goal = Noneself.min_rand = randArea[0]self.max_rand = randArea[1]self.expand_dis = expandDisself.goal_sample_rate = goalSampleRateself.max_iter = maxIterself.obstacle_list = obstacleList# 存储RRT树self.node_list = None# start、goal 起点终点坐标def rrt_planning(self, start, goal, animation=True):start_time = time.time()self.start = Node(start[0], start[1])self.goal = Node(goal[0], goal[1])# 将起点加入node_list作为树的根结点self.node_list = [self.start]path = Nonefor i in range(self.max_iter):# 进行采样rnd = self.sample()# 取的距离采样点最近的节点下标n_ind = self.get_nearest_list_index(self.node_list, rnd)# 得到最近节点nearestNode = self.node_list[n_ind]# 将Xrandom和Xnear连线方向作为生长方向# math.atan2() 函数接受两个参数，分别是 y 坐标差值和 x 坐标差值。它返回的值是以弧度表示的角度，范围在 -π 到 π 之间。这个角度表示了从 nearestNode 指向 rnd 的方向。theta = math.atan2(rnd[1] - nearestNode.y, rnd[0] - nearestNode.x)# 生长 ： 输入参数为角度、下标、nodelist中最近的节点  得到生长过后的节点newNode = self.get_new_node(theta, n_ind, nearestNode)# 检查是否有障碍物 传入参数为新生城路径的两个节点noCollision = self.check_segment_collision(newNode.x, newNode.y, nearestNode.x, nearestNode.y)if noCollision:# 没有碰撞把新节点加入到树里面self.node_list.append(newNode)if animation:self.draw_graph(newNode, path)# 是否到终点附近if self.is_near_goal(newNode):# 是否这条路径与障碍物发生碰撞if self.check_segment_collision(newNode.x, newNode.y,self.goal.x, self.goal.y):lastIndex = len(self.node_list) - 1# 找路径path = self.get_final_course(lastIndex)pathLen = self.get_path_len(path)print("current path length: {}, It costs {} s".format(pathLen, time.time()-start_time))if animation:self.draw_graph(newNode, path)return pathdef rrt_star_planning(self, start, goal, animation=True):start_time = time.time()self.start = Node(start[0], start[1])self.goal = Node(goal[0], goal[1])self.node_list = [self.start]path = NonelastPathLength = float('inf')for i in range(self.max_iter):rnd = self.sample()n_ind = self.get_nearest_list_index(self.node_list, rnd)nearestNode = self.node_list[n_ind]# steertheta = math.atan2(rnd[1] - nearestNode.y, rnd[0] - nearestNode.x)newNode = self.get_new_node(theta, n_ind, nearestNode)noCollision = self.check_segment_collision(newNode.x, newNode.y, nearestNode.x, nearestNode.y)if noCollision:nearInds = self.find_near_nodes(newNode)newNode = self.choose_parent(newNode, nearInds)self.node_list.append(newNode)# 重联self.rewire(newNode, nearInds)if animation:self.draw_graph(newNode, path)if self.is_near_goal(newNode):if self.check_segment_collision(newNode.x, newNode.y,self.goal.x, self.goal.y):lastIndex = len(self.node_list) - 1tempPath = self.get_final_course(lastIndex)tempPathLen = self.get_path_len(tempPath)if lastPathLength > tempPathLen:path = tempPathlastPathLength = tempPathLenprint("current path length: {}, It costs {} s".format(tempPathLen, time.time()-start_time))return pathdef informed_rrt_star_planning(self, start, goal, animation=True):start_time = time.time()self.start = Node(start[0], start[1])self.goal = Node(goal[0], goal[1])self.node_list = [self.start]# max length we expect to find in our 'informed' sample space,# starts as infinitecBest = float('inf')path = None# Computing the sampling spacecMin = math.sqrt(pow(self.start.x - self.goal.x, 2)+ pow(self.start.y - self.goal.y, 2))xCenter = np.array([[(self.start.x + self.goal.x) / 2.0],[(self.start.y + self.goal.y) / 2.0], [0]])a1 = np.array([[(self.goal.x - self.start.x) / cMin],[(self.goal.y - self.start.y) / cMin], [0]])e_theta = math.atan2(a1[1], a1[0])# 论文方法求旋转矩阵（2选1）# first column of identity matrix transposed# id1_t = np.array([1.0, 0.0, 0.0]).reshape(1, 3)# M = a1 @ id1_t# U, S, Vh = np.linalg.svd(M, True, True)# C = np.dot(np.dot(U, np.diag(#     [1.0, 1.0, np.linalg.det(U) * np.linalg.det(np.transpose(Vh))])),#            Vh)# 直接用二维平面上的公式（2选1）C = np.array([[math.cos(e_theta), -math.sin(e_theta), 0],[math.sin(e_theta), math.cos(e_theta),  0],[0,                 0,                  1]])for i in range(self.max_iter):# Sample space is defined by cBest# cMin is the minimum distance between the start point and the goal# xCenter is the midpoint between the start and the goal# cBest changes when a new path is foundrnd = self.informed_sample(cBest, cMin, xCenter, C)n_ind = self.get_nearest_list_index(self.node_list, rnd)nearestNode = self.node_list[n_ind]# steertheta = math.atan2(rnd[1] - nearestNode.y, rnd[0] - nearestNode.x)newNode = self.get_new_node(theta, n_ind, nearestNode)noCollision = self.check_segment_collision(newNode.x, newNode.y, nearestNode.x, nearestNode.y)if noCollision:nearInds = self.find_near_nodes(newNode)newNode = self.choose_parent(newNode, nearInds)self.node_list.append(newNode)self.rewire(newNode, nearInds)if self.is_near_goal(newNode):if self.check_segment_collision(newNode.x, newNode.y,self.goal.x, self.goal.y):lastIndex = len(self.node_list) - 1tempPath = self.get_final_course(lastIndex)tempPathLen = self.get_path_len(tempPath)if tempPathLen < cBest:path = tempPathcBest = tempPathLenprint("current path length: {}, It costs {} s".format(tempPathLen, time.time()-start_time))if animation:self.draw_graph_informed_RRTStar(xCenter=xCenter,cBest=cBest, cMin=cMin,e_theta=e_theta, rnd=rnd, path=path)return pathdef sample(self):# 取得1-100的随机数,如果比10大的话（以10%的概率取到终点）if random.randint(0, 100) > self.goal_sample_rate:# 在空间里随机采样一个点rnd = [random.uniform(self.min_rand, self.max_rand), random.uniform(self.min_rand, self.max_rand)]else:  # goal point sampling# 终点作为采样点rnd = [self.goal.x, self.goal.y]return rnddef choose_parent(self, newNode, nearInds):# 圈里是否有候选节点if len(nearInds) == 0:return newNodedList = []for i in nearInds:dx = newNode.x - self.node_list[i].xdy = newNode.y - self.node_list[i].yd = math.hypot(dx, dy)theta = math.atan2(dy, dx)# 检测是否碰到障碍物if self.check_collision(self.node_list[i], theta, d):# 计算距离dList.append(self.node_list[i].cost + d)else:# 无穷大dList.append(float('inf'))# 找到路径最小的的点 父节点minCost = min(dList)minInd = nearInds[dList.index(minCost)]if minCost == float('inf'):print("min cost is inf")return newNodenewNode.cost = minCostnewNode.parent = minIndreturn newNodedef find_near_nodes(self, newNode):n_node = len(self.node_list)# 动态变化 搜索到越后面的话半径越小r = 50.0 * math.sqrt((math.log(n_node) / n_node))# 找到节点d_list = [(node.x - newNode.x) ** 2 + (node.y - newNode.y) ** 2for node in self.node_list]# 比半径小near_inds = [d_list.index(i) for i in d_list if i <= r ** 2]return near_indsdef informed_sample(self, cMax, cMin, xCenter, C):if cMax < float('inf'):r = [cMax / 2.0,math.sqrt(cMax ** 2 - cMin ** 2) / 2.0,math.sqrt(cMax ** 2 - cMin ** 2) / 2.0]L = np.diag(r)xBall = self.sample_unit_ball()rnd = np.dot(np.dot(C, L), xBall) + xCenterrnd = [rnd[(0, 0)], rnd[(1, 0)]]else:rnd = self.sample()return rnd@staticmethoddef sample_unit_ball():a = random.random()b = random.random()if b < a:a, b = b, asample = (b * math.cos(2 * math.pi * a / b),b * math.sin(2 * math.pi * a / b))return np.array([[sample[0]], [sample[1]], [0]])@staticmethoddef get_path_len(path):pathLen = 0for i in range(1, len(path)):node1_x = path[i][0]node1_y = path[i][1]node2_x = path[i - 1][0]node2_y = path[i - 1][1]pathLen += math.sqrt((node1_x - node2_x)** 2 + (node1_y - node2_y) ** 2)return pathLen@staticmethoddef line_cost(node1, node2):return math.sqrt((node1.x - node2.x) ** 2 + (node1.y - node2.y) ** 2)@staticmethoddef get_nearest_list_index(nodes, rnd):# 遍历所有节点 计算采样点和节点的距离dList = [(node.x - rnd[0]) ** 2+ (node.y - rnd[1]) ** 2 for node in nodes]# 获得最近的距离所对应的索引minIndex = dList.index(min(dList))return minIndexdef get_new_node(self, theta, n_ind, nearestNode):newNode = copy.deepcopy(nearestNode)# 坐标newNode.x += self.expand_dis * math.cos(theta)newNode.y += self.expand_dis * math.sin(theta)# 代价newNode.cost += self.expand_dis# 父亲节点newNode.parent = n_indreturn newNodedef is_near_goal(self, node):# 计算距离d = self.line_cost(node, self.goal)if d < self.expand_dis:return Truereturn Falsedef rewire(self, newNode, nearInds):n_node = len(self.node_list)# 新节点 和圆圈的候选点for i in nearInds:nearNode = self.node_list[i]# 以newnode作为父节点 计算两条节点之间的距离d = math.sqrt((nearNode.x - newNode.x) ** 2+ (nearNode.y - newNode.y) ** 2)s_cost = newNode.cost + dif nearNode.cost > s_cost:theta = math.atan2(newNode.y - nearNode.y,newNode.x - nearNode.x)if self.check_collision(nearNode, theta, d):nearNode.parent = n_node - 1nearNode.cost = s_cost@staticmethoddef distance_squared_point_to_segment(v, w, p):# Return minimum distance between line segment vw and point pif np.array_equal(v, w):return (p - v).dot(p - v)  # v == w casel2 = (w - v).dot(w - v)  # i.e. |w-v|^2 -  avoid a sqrt# Consider the line extending the segment,# parameterized as v + t (w - v).# We find projection of point p onto the line.# It falls where t = [(p-v) . (w-v)] / |w-v|^2# We clamp t from [0,1] to handle points outside the segment vw.t = max(0, min(1, (p - v).dot(w - v) / l2))projection = v + t * (w - v)  # Projection falls on the segmentreturn (p - projection).dot(p - projection)def check_segment_collision(self, x1, y1, x2, y2):# 遍历所有的障碍物for (ox, oy, size) in self.obstacle_list:dd = self.distance_squared_point_to_segment(np.array([x1, y1]),np.array([x2, y2]),np.array([ox, oy]))if dd <= size ** 2:return False  # collisionreturn Truedef check_collision(self, nearNode, theta, d):tmpNode = copy.deepcopy(nearNode)end_x = tmpNode.x + math.cos(theta) * dend_y = tmpNode.y + math.sin(theta) * dreturn self.check_segment_collision(tmpNode.x, tmpNode.y, end_x, end_y)def get_final_course(self, lastIndex):path = [[self.goal.x, self.goal.y]]while self.node_list[lastIndex].parent is not None:node = self.node_list[lastIndex]path.append([node.x, node.y])lastIndex = node.parentpath.append([self.start.x, self.start.y])return pathdef draw_graph_informed_RRTStar(self, xCenter=None, cBest=None, cMin=None, e_theta=None, rnd=None, path=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])if rnd is not None:plt.plot(rnd[0], rnd[1], "^k")if cBest != float('inf'):self.plot_ellipse(xCenter, cBest, cMin, e_theta)for node in self.node_list:if node.parent is not None:if node.x or node.y is not None:plt.plot([node.x, self.node_list[node.parent].x], [node.y, self.node_list[node.parent].y], "-g")for (ox, oy, size) in self.obstacle_list:plt.plot(ox, oy, "ok", ms=30 * size)if path is not None:plt.plot([x for (x, y) in path], [y for (x, y) in path], '-r')plt.plot(self.start.x, self.start.y, "xr")plt.plot(self.goal.x, self.goal.y, "xr")plt.axis([-2, 18, -2, 15])plt.grid(True)plt.pause(0.01)@staticmethoddef plot_ellipse(xCenter, cBest, cMin, e_theta):  # pragma: no covera = math.sqrt(cBest ** 2 - cMin ** 2) / 2.0b = cBest / 2.0angle = math.pi / 2.0 - e_thetacx = xCenter[0]cy = xCenter[1]t = np.arange(0, 2 * math.pi + 0.1, 0.1)x = [a * math.cos(it) for it in t]y = [b * math.sin(it) for it in t]rot = Rot.from_euler('z', -angle).as_matrix()[0:2, 0:2]fx = rot @ np.array([x, y])px = np.array(fx[0, :] + cx).flatten()py = np.array(fx[1, :] + cy).flatten()plt.plot(cx, cy, "xc")plt.plot(px, py, "--c")def draw_graph(self, rnd=None, path=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])if rnd is not None:plt.plot(rnd.x, rnd.y, "^k")for node in self.node_list:if node.parent is not None:if node.x or node.y is not None:plt.plot([node.x, self.node_list[node.parent].x], [node.y, self.node_list[node.parent].y], "-g")for (ox, oy, size) in self.obstacle_list:# self.plot_circle(ox, oy, size)plt.plot(ox, oy, "ok", ms=30 * size)plt.plot(self.start.x, self.start.y, "xr")plt.plot(self.goal.x, self.goal.y, "xr")if path is not None:plt.plot([x for (x, y) in path], [y for (x, y) in path], '-r')plt.axis([-2, 18, -2, 15])plt.grid(True)plt.pause(0.01)class Node:def __init__(self, x, y):self.x = xself.y = yself.cost = 0.0self.parent = Nonedef main():print("Start rrt planning")# create obstacles# obstacleList = [#     (3,  3,  1.5),#     (12, 2,  3),#     (3,  9,  2),#     (9,  11, 2),# ]# 设置障碍物 (圆点、半径)obstacleList = [(5, 5, 1), (3, 6, 2), (3, 8, 2), (3, 10, 2), (7, 5, 2),(9, 5, 2), (8, 10, 1)]# Set params# 采样范围 设置的障碍物 最大迭代次数rrt = RRT(randArea=[-2, 18], obstacleList=obstacleList, maxIter=200)# 传入的是起点和终点#path = rrt.rrt_planning(start=[0, 0], goal=[15, 12], animation=show_animation)#path = rrt.rrt_star_planning(start=[0, 0], goal=[15, 12], animation=show_animation)path = rrt.informed_rrt_star_planning(start=[0, 0], goal=[15, 12], animation=show_animation)print("Done!!")if show_animation and path:plt.show()if __name__ == '__main__':main()