在上一节中，介绍了 RRT 算法的原理，这一节将一步步实现 RRT 路径规划算法在二维环境中的路径规划，来进一步加深对 RRT 算法的理解。
二维环境的搭建
我们将搭建下图所示的二维环境，绿色点为起点（0，0），红色点为目标点（15, 12），黑色的圆表示障碍物。
实现上述环境的代码如下：
start = [0, 0]# 起点goal = [15, 12]# 终点# 障碍物 （x, y, radiu）obstacle_list = [(3, 3, 1.5),(12, 2, 3),(3, 9, 2),(9, 11, 2)]plt.axis([-2, 18, -2, 15])for (ox, oy, size) in obstacle_list:plt.plot(ox, oy, "ok", ms=30 * size)plt.plot(start[0], start[1], "og")plt.plot(goal[0], goal[1], "or")plt.show()plt.pause(0.01)
RRT 路径规划算法
算法初始化
class RRT:# 初始化def __init__(self, obstacle_list, # 障碍物rand_area, # 采样的区域expand_dis=2.0, # 步长goal_sample_rate=10, # 目标采样率max_iter=200): # 最大迭代次数self.start = Noneself.goal = Noneself.min_rand = rand_area[0]self.max_rand = rand_area[1]self.expand_dis = expand_disself.goal_sample_rate = goal_sample_rateself.max_iter = max_iterself.obstacle_list = obstacle_listself.node_list = None
路径规划
将起点和终点结点化，方便计算该结点到起点的路径距离以及后面的路径回溯。
def rrt_planning(self, start, goal, animation=True):self.start = Node(start[0], start[1])self.goal = Node(goal[0], goal[1])self.node_list[self.start]path = None
结点化的代码如下：
class Node: def __init__(self, x, y):self.x = xself.y = yself.cost = 0.0self.parent = None
开始在环境中循环采样点，在此有一个小的采样技巧，为了使 RRT 树能朝着目标点的方向生长，在采样时，以一定的概率采样目标点。
rnd = self.sample() # 在环境中随机采样点
采样函数如下：
def sample(self):if random.randint(0, 100) > self.goal_sample_rate:rnd = [random.uniform(self.min_rand, self.max_rand), random.uniform(self.min_rand, self.max_rand)]else:rnd = [self.goal.x, self.goal.y]return rnd
从结点树中找到距离采样点最近的结点。
n_ind = self.get_nearest_list_index(self.node_list, rnd)nearest_node = self.node_list[n_ind]
def get_nearest_list_index(nodes, rnd):""" 计算树中距离采样点距离最近的结点 """d_list = [(node.x - rnd[0]) ** 2 + (node.y - rnd[1]) ** 2 for node in nodes]min_index = d_list.index(min(d_list))return min_index
生成新的下一个结点。在找到树中距离采样点最近的结点后，可以计算两个结点的连线和水平的方向的角度，再根据步长的大小，即可计算出下一个树结点的位置。
theta = math.atan2(rnd[1] - nearest_node.y, rnd[0] - nearest_node.x)new_node = self.get_new_node(theta, n_ind, nearest_node)
def get_new_node(self, theta, n_ind, nearest_node):""" 计算新结点 """new_node = copy.deepcopy(nearest_node)new_node.x += self.expand_dis * math.cos(theta)new_node.y += self.expand_dis * math.sin(theta)new_node.cost += self.expand_disnew_node.parent = n_indreturn new_node
检测碰撞。检测新生成的结点的路径是否会与障碍物碰撞
no_collision = self.check_segment_collision(new_node.x, new_node.y, nearest_node.x, nearest_node.y)
其中检测碰撞的函数如下：
def check_segment_collision(self, x1, y1, x2, y2):for (ox, oy, radius) 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 <= radius ** 2:return Falsereturn True
其中distance_squared_point_to_segment()函数的功能为：求点到线段的最短距离，代码如下：
def distance_squared_point_to_segment(v, w, p):""" 计算线段 vw 和 点 p 之间的最短距离"""if np.array_equal(v, w): # 点 v 和 点 w 重合的情况return (p - v).dot(p - v)l2 = (w - v).dot(w - v)# 线段 vw 长度的平方t = max(0, min(1, (p - v).dot(w - v) / l2))projection = v + t * (w - v)return (p - projection).dot(p - projection)
如果没有与障碍物发生碰撞，则将新结点加入到树中，并绘制新结点以及生长的新枝干。再判断新结点是否是目标点的邻接结点。
if no_collision:self.node_list.append(new_node)# 一步一绘制if animation:self.draw_graph(new_node, path)# 判断新结点是否临近目标点if self.is_near_goal(new_node):if self.check_segment_collision(new_node.x, new_node.y,self.goal.x, self.goal.y):last_index = len(self.node_list) - 1path = self.get_final_course(last_index) # 回溯路径path_length = self.get_path_len(path) # 计算路径的长度print("当前的路径长度为：{}".format(path_length))if animation:self.draw_graph(new_node, path)return path
其中，is_near_goal()是判断新结点是否邻近目标点的函数，其代码如下：
def is_near_goal(self, node):d = self.line_cost(node, self.goal)if d < self.expand_dis:return Truereturn False
line_cost()函数是如果新生成的结点邻近目标结点的情况下，该结点到目标结点之间的距离。其代码如下：
def line_cost(node1, node2):return math.sqrt((node1.x - node2.x) ** 2 + (node1.y - node2.y) ** 2)
get_final_course()是获取最终从起点到终点的路径的函数。其代码如下：
def get_final_course(self, last_index):""" 回溯路径 """path = [[self.goal.x, self.goal.y]]while self.node_list[last_index].parent is not None:node = self.node_list[last_index]path.append([node.x, node.y])last_index = node.parentpath.append([self.start.x, self.start.y])return path
get_path_len()是求取路径的总长度的函数，其代码如下：
def get_path_len(path):""" 计算路径的长度 """path_length = 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]path_length += math.sqrt((node1_x - node2_x) ** 2 + (node1_y - node2_y) ** 2)return path_length
draw_graph()为绘制地图以及结点路径等函数，使之可视化。其代码如下：
def draw_graph(self, rnd=None, path=None):plt.clf()# 绘制新的结点if rnd is not None:plt.plot(rnd.x, rnd.y, '^k')# 绘制路径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')# 绘制障碍物for (ox, oy, size) in self.obstacle_list:plt.plot(ox, oy, "ok", ms=30 * size)# 绘制起点、终点plt.plot(self.start.x, self.start.y, "og")plt.plot(self.goal.x, self.goal.y, "or")# 绘制路径if path is not None:plt.plot([x for (x, y) in path], [y for (x, y) in path], '-r')# 绘制图的设置plt.axis([-2, 18, -2, 15])plt.grid(True)plt.pause(0.01)
最终运行结果如下：
完整代码如下：
import randomimport mathimport copyimport timeimport matplotlib.pyplot as pltimport numpy as npclass RRT:# 初始化def __init__(self,obstacle_list, # 障碍物rand_area, # 采样的区域expand_dis=2.0, # 步长goal_sample_rate=10, # 目标采样率max_iter=200): # 最大迭代次数self.start = Noneself.goal = Noneself.min_rand = rand_area[0]self.max_rand = rand_area[1]self.expand_dis = expand_disself.goal_sample_rate = goal_sample_rateself.max_iter = max_iterself.obstacle_list = obstacle_listself.node_list = Nonedef rrt_planning(self, start, goal, animation=True):self.start = Node(start[0], start[1])self.goal = Node(goal[0], goal[1])self.node_list = [self.start]path = Nonefor i in range(self.max_iter):# 1. 在环境中随机采样点rnd = self.sample()# 2. 找到结点树中距离采样点最近的结点n_ind = self.get_nearest_list_index(self.node_list, rnd)nearest_node = self.node_list[n_ind]# 3. 在采样点的方向生长一个步长，得到下一个树的结点。theta = math.atan2(rnd[1] - nearest_node.y, rnd[0] - nearest_node.x)new_node = self.get_new_node(theta, n_ind, nearest_node)# 4. 检测碰撞，检测到新生成的结点的路径是否会与障碍物碰撞no_collision = self.check_segment_collision(new_node.x, new_node.y, nearest_node.x, nearest_node.y)if no_collision:self.node_list.append(new_node)# 一步一绘制if animation:time.sleep(1)self.draw_graph(new_node, path)# 判断新结点是否临近目标点if self.is_near_goal(new_node):if self.check_segment_collision(new_node.x, new_node.y,self.goal.x, self.goal.y):last_index = len(self.node_list) - 1path = self.get_final_course(last_index) # 回溯路径path_length = self.get_path_len(path) # 计算路径的长度print("当前的路径长度为：{}".format(path_length))if animation:self.draw_graph(new_node, path)return pathdef sample(self):""" 在环境中采样点的函数，以一定的概率采样目标点 """if random.randint(0, 100) > self.goal_sample_rate:rnd = [random.uniform(self.min_rand, self.max_rand),random.uniform(self.min_rand, self.max_rand)]else:rnd = [self.goal.x, self.goal.y]return rnd@staticmethoddef get_nearest_list_index(nodes, rnd):""" 计算树中距离采样点距离最近的结点 """d_list = [(node.x - rnd[0]) ** 2 + (node.y - rnd[1]) ** 2for node in nodes]min_index = d_list.index(min(d_list))return min_indexdef get_new_node(self, theta, n_ind, nearest_node):""" 计算新结点 """new_node = copy.deepcopy(nearest_node)new_node.x += self.expand_dis * math.cos(theta)new_node.y += self.expand_dis * math.sin(theta)new_node.cost += self.expand_disnew_node.parent = n_indreturn new_nodedef check_segment_collision(self, x1, y1, x2, y2):""" 检测碰撞 """for (ox, oy, radius) 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 <= radius ** 2:return Falsereturn True@staticmethoddef distance_squared_point_to_segment(v, w, p):""" 计算线段 vw 和 点 p 之间的最短距离"""if np.array_equal(v, w): # 点 v 和 点 w 重合的情况return (p - v).dot(p - v)l2 = (w - v).dot(w - v)# 线段 vw 长度的平方t = max(0, min(1, (p - v).dot(w - v) / l2))projection = v + t * (w - v)return (p - projection).dot(p - projection)def draw_graph(self, rnd=None, path=None):plt.clf()# 绘制新的结点if rnd is not None:plt.plot(rnd.x, rnd.y, '^k')# 绘制路径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')# 绘制起点、终点plt.plot(self.start.x, self.start.y, "og")plt.plot(self.goal.x, self.goal.y, "or")# 绘制障碍物for (ox, oy, size) in self.obstacle_list:plt.plot(ox, oy, "ok", ms=30 * size)# 绘制路径if path is not None:plt.plot([x for (x, y) in path], [y for (x, y) in path], '-r')# 绘制图的设置plt.axis([-2, 18, -2, 15])plt.grid(True)plt.pause(0.01)def is_near_goal(self, node):d = self.line_cost(node, self.goal)if d < self.expand_dis:return Truereturn False@staticmethoddef line_cost(node1, node2):return math.sqrt((node1.x - node2.x) ** 2 + (node1.y - node2.y) ** 2)def get_final_course(self, last_index):""" 回溯路径 """path = [[self.goal.x, self.goal.y]]while self.node_list[last_index].parent is not None:node = self.node_list[last_index]path.append([node.x, node.y])last_index = node.parentpath.append([self.start.x, self.start.y])return path@staticmethoddef get_path_len(path):""" 计算路径的长度 """path_length = 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]path_length += math.sqrt((node1_x - node2_x) ** 2 + (node1_y - node2_y) ** 2)return path_lengthclass Node:def __init__(self, x, y):self.x = xself.y = yself.cost = 0.0self.parent = Nonedef main():print('Start RRT planning!')show_animation = Truestart = [0, 0]goal = [15, 12]# 障碍物 (x, y, radius)obstacle_list = [(3, 3, 1.5),(12, 2, 3),(3, 9, 2),(9, 11, 2)]rrt = RRT(rand_area=[-2, 18], obstacle_list=obstacle_list, max_iter=200)path = rrt.rrt_planning(start=[0, 0], goal=[15, 12], animation=show_animation)print('Done!')if show_animation and path:plt.show()if __name__ == '__main__':main()
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