s什么网站可以接单做设计赚钱,做建网站,建站软件公司,张家港做网站广告公司目录 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#xff1a;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#xff1f;…目录 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,expandDis2.0, goalSampleRate10, maxIter200):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, animationTrue):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, animationTrue):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, animationTrue):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1C 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(xCenterxCenter,cBestcBest, cMincMin,e_thetae_theta, rndrnd, pathpath)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 rndstaticmethoddef 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 pathLenstaticmethoddef 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_coststaticmethoddef 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, xCenterNone, cBestNone, cMinNone, e_thetaNone, rndNone, pathNone):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, ms30 * 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, rndNone, pathNone):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, ms30 * 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], obstacleListobstacleList, maxIter200)# 传入的是起点和终点#path rrt.rrt_planning(start[0, 0], goal[15, 12], animationshow_animation)#path rrt.rrt_star_planning(start[0, 0], goal[15, 12], animationshow_animation)path rrt.informed_rrt_star_planning(start[0, 0], goal[15, 12], animationshow_animation)print(Done!!)if show_animation and path:plt.show()if __name__ __main__:main()4 算法比较 RRT路径较差非最优在本例程中为6ms。 RRT*路径渐进最优在本例程中为34ms。 Informed RRT*路径渐进最优在本例程中为109ms。 看官姥爷看自己需求用奥。
