一文教你用python编写Dijkstra算法进行机器人路径规划一文教你用python编写Dijkstra算法

前言
一、算法原理
二、程序代码
三、运行结果
四、 A*算法：Djikstra算法的改进
总结

前言为了机器人在寻路的过程中避障并且找到最短距离，我们需要使用一些算法进行路径规划（Path Planning），常用的算法有Djikstra算法、A*算法等等，在github上有一个非常好的项目叫做PythonRobotics，其中给出了源代码，参考代码，可以对Djikstra算法有更深的了解。

一、算法原理

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如图所示，Dijkstra算法要解决的是一个有向权重图中最短路径的寻找问题，图中红色节点1代表起始节点，蓝色节点6代表目标结点。箭头上的数字代表两个结点中的的距离，也就是模型中所谓的代价（cost）。
贪心算法需要设立两个集合，open_set（开集）和closed_set（闭集），然后根据以下程序进行操作：

把初始结点放入到open_set中；
把open_set中代价最小的节点取出来放入到closed_set中，并且作为当前节点；
把与当前节点相邻的节点放入到open_set中，如果代价更小更新代价
重复2-3过程，直到找到终点。

注意open_set中的代价是可变的，而closed_set中的代价已经是最小的代价了，这也是为什么叫做open和close的原因。
至于为什么closed_set中的代价是最小的，是因为我们使用了贪心算法，既然已经把节点加入到了close中，那么初始点到close节点中的距离就比到open中的距离小了，无论如何也不可能找到比它更小的了。

二、程序代码

"""Grid based Dijkstra planningauthor: Atsushi Sakai(@Atsushi_twi)"""import matplotlib.pyplot as pltimport mathshow_animation = Trueclass Dijkstra:def __init__(self, ox, oy, resolution, robot_radius):"""Initialize map for a star planningox: x position list of Obstacles [m]oy: y position list of Obstacles [m]resolution: grid resolution [m]rr: robot radius[m]"""self.min_x = Noneself.min_y = Noneself.max_x = Noneself.max_y = Noneself.x_width = Noneself.y_width = Noneself.obstacle_map = Noneself.resolution = resolutionself.robot_radius = robot_radiusself.calc_obstacle_map(ox, oy)self.motion = self.get_motion_model()class Node:def __init__(self, x, y, cost, parent_index):self.x = x# index of gridself.y = y# index of gridself.cost = costself.parent_index = parent_index# index of previous Nodedef __str__(self):return str(self.x) + "," + str(self.y) + "," + str(self.cost) + "," + str(self.parent_index)def planning(self, sx, sy, gx, gy):"""dijkstra path searchinput:s_x: start x position [m]s_y: start y position [m]gx: goal x position [m]gx: goal x position [m]output:rx: x position list of the final pathry: y position list of the final path"""start_node = self.Node(self.calc_xy_index(sx, self.min_x),self.calc_xy_index(sy, self.min_y), 0.0, -1)goal_node = self.Node(self.calc_xy_index(gx, self.min_x),self.calc_xy_index(gy, self.min_y), 0.0, -1)open_set, closed_set = dict(), dict()open_set[self.calc_index(start_node)] = start_nodewhile 1:c_id = min(open_set, key=lambda o: open_set[o].cost)current = open_set[c_id]# show graphif show_animation:# pragma: no coverplt.plot(self.calc_position(current.x, self.min_x),self.calc_position(current.y, self.min_y), "xc")# 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 len(closed_set.keys()) % 10 == 0:plt.pause(0.001)if current.x == goal_node.x and current.y == goal_node.y:print("Find goal")goal_node.parent_index = current.parent_indexgoal_node.cost = current.costbreak# Remove the item from the open setdel open_set[c_id]# Add it to the closed setclosed_set[c_id] = current# expand search grid based on motion modelfor move_x, move_y, move_cost in self.motion:node = self.Node(current.x + move_x,current.y + move_y,current.cost + move_cost, c_id)n_id = self.calc_index(node)if n_id in closed_set:continueif not self.verify_node(node):continueif n_id not in open_set:open_set[n_id] = node# Discover a new nodeelse:if open_set[n_id].cost >= node.cost:# This path is the best until now. record it!open_set[n_id] = noderx, ry = self.calc_final_path(goal_node, closed_set)return rx, rydef calc_final_path(self, goal_node, closed_set):# generate final courserx, ry = [self.calc_position(goal_node.x, self.min_x)], [self.calc_position(goal_node.y, self.min_y)]parent_index = goal_node.parent_indexwhile parent_index != -1:n = closed_set[parent_index]rx.append(self.calc_position(n.x, self.min_x))ry.append(self.calc_position(n.y, self.min_y))parent_index = n.parent_indexreturn rx, rydef calc_position(self, index, minp):pos = index * self.resolution + minpreturn posdef calc_xy_index(self, position, minp):return round((position - minp) / self.resolution)def calc_index(self, node):return (node.y - self.min_y) * self.x_width + (node.x - self.min_x)def verify_node(self, node):px = self.calc_position(node.x, self.min_x)py = self.calc_position(node.y, self.min_y)if px < self.min_x:return Falseif py < self.min_y:return Falseif px >= self.max_x:return Falseif py >= self.max_y:return Falseif self.obstacle_map[node.x][node.y]:return Falsereturn Truedef calc_obstacle_map(self, ox, oy):self.min_x = round(min(ox))self.min_y = round(min(oy))self.max_x = round(max(ox))self.max_y = round(max(oy))print("min_x:", self.min_x)print("min_y:", self.min_y)print("max_x:", self.max_x)print("max_y:", self.max_y)self.x_width = round((self.max_x - self.min_x) / self.resolution)self.y_width = round((self.max_y - self.min_y) / self.resolution)print("x_width:", self.x_width)print("y_width:", self.y_width)# obstacle map generationself.obstacle_map = [[False for _ in range(self.y_width)]for _ in range(self.x_width)]for ix in range(self.x_width):x = self.calc_position(ix, self.min_x)for iy in range(self.y_width):y = self.calc_position(iy, self.min_y)for iox, ioy in zip(ox, oy):d = math.hypot(iox - x, ioy - y)if d <= self.robot_radius:self.obstacle_map[ix][iy] = Truebreak@staticmethoddef get_motion_model():# dx, dy, costmotion = [[1, 0, 1],[0, 1, 1],[-1, 0, 1],[0, -1, 1],[-1, -1, math.sqrt(2)],[-1, 1, math.sqrt(2)],[1, -1, math.sqrt(2)],[1, 1, math.sqrt(2)]]return motiondef main():print(__file__ + " start!!")# start and goal positionsx = -5.0# [m]sy = -5.0# [m]gx = 50.0# [m]gy = 50.0# [m]grid_size = 2.0# [m]robot_radius = 1.0# [m]# set obstacle positionsox, oy = [], []for i in range(-10, 60):ox.append(i)oy.append(-10.0)for i in range(-10, 60):ox.append(60.0)oy.append(i)for i in range(-10, 61):ox.append(i)oy.append(60.0)for i in range(-10, 61):ox.append(-10.0)oy.append(i)for i in range(-10, 40):ox.append(20.0)oy.append(i)for i in range(0, 40):ox.append(40.0)oy.append(60.0 - i)if show_animation:# pragma: no coverplt.plot(ox, oy, ".k")plt.plot(sx, sy, "og")plt.plot(gx, gy, "xb")plt.grid(True)plt.axis("equal")dijkstra = Dijkstra(ox, oy, grid_size, robot_radius)rx, ry = dijkstra.planning(sx, sy, gx, gy)if show_animation:# pragma: no coverplt.plot(rx, ry, "-r")plt.pause(0.01)plt.show()if __name__ == '__main__':main()

三、运行结果

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四、 A*算法：Djikstra算法的改进 Dijkstra算法实际上是贪心搜索算法，算法复杂度为O( n 2 n^2 n2)，为了减少无效搜索的次数，我们可以增加一个启发式函数（heuristic），比如搜索点到终点目标的距离，在选择open_set元素的时候，我们将cost变成cost+heuristic，就可以给出搜索的方向性，这样就可以减少南辕北辙的情况。我们可以run一下PythonRobotics中的Astar代码，得到以下结果：

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