Dijkstra算法本身是基于贪心，广度优先，动态优化计算最短距离，最好事先了解一下Dijkstra算法本质上是什么，然后再看在路径规划上的应用比较好。这里贴出来Dijkstra算法介绍的知乎文章Dijkstra算法详解 通俗易懂 - 知乎 (zhihu.com)，这里就不做讲解。
示例如下
伪代码
Python代码
"""
Grid based Dijkstra planning
author: Atsushi Sakai(@Atsushi_twi)
"""
import matplotlib.pyplot as plt
import math
show_animation = True
class Dijkstra:
def __init__(self, ox, oy, resolution, robot_radius):
"""
Initialize map for planning
ox: x position list of Obstacles [m]
oy: y position list of Obstacles [m]
resolution: grid resolution [m]
rr: robot radius[m]
"""
self.min_x = None
self.min_y = None
self.max_x = None
self.max_y = None
self.x_width = None
self.y_width = None
self.obstacle_map = None
# 栅格的大小
self.resolution = resolution
# 机器人的半径
self.robot_radius = robot_radius
# 构建栅格地图,传入的是障碍物信息
self.calc_obstacle_map(ox, oy)
# 机器人的运动方式
self.motion = self.get_motion_model()
class Node:
def __init__(self, x, y, cost, parent_index):
self.x = x # index of grid
self.y = y # index of grid
self.cost = cost # g(n)
self.parent_index = parent_index # index of previous Node
def __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 search
input:
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 path
ry: y position list of the final path
"""
# 求的就是X方向的索引,也就是说在栅格地图中的相对位置
start_node = self.Node(self.calc_xy_index(sx, self.min_x),
self.calc_xy_index(sy, self.min_y), 0.0,
-1) # round((position - minp) / self.resolution)
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() # key - value: hash表
# 这里需要理解一下就是我们将栅格地图中的每一个位置都编一个号,从第零行,y=0~y=最大值,一直到最大行,例如下面的例子
'''
13 14 15 16 17...
7 8 9 10 11 12
1 2 3 4 5 6
'''
# 因为open_set是字典结构,所以需要键值对,
# self.calc_index(start_node)就是起始点的索引
# 对应的值就是start_node,这样就形成了键值对
open_set[self.calc_index(start_node)] = start_node
# 进入循环
while 1:
# 取cost最小的节点 key=lambda这个是匿名函数的意思,
# o: open_set[o].cost就是说按照这个来进行排序
# 取cost最小的节点的索引
c_id = min(open_set, key=lambda o: open_set[o].cost)
# 从open_set中把这个节点取出来
current = open_set[c_id]
# show graph
if show_animation: # pragma: no cover
plt.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)
# 判断是否是终点
if current.x == goal_node.x and current.y == goal_node.y:
print("Find goal")
goal_node.parent_index = current.parent_index
goal_node.cost = current.cost
# 如果是就退出循环
break
# 把这个节点从open_set中删除
del open_set[c_id]
# 把该点加入到closed_set中
closed_set[c_id] = current
# 遍历该点的邻接节点,通过运动方式进行遍历
for move_x, move_y, move_cost in self.motion:
# 求得邻接节点的x,y,cost和父节点
node = self.Node(current.x + move_x,
current.y + move_y,
current.cost + move_cost, c_id)
# 计算对应的索引值
n_id = self.calc_index(node)
# 判断这个索引是否已经在closed_set里了，也就是是否已经在路径中了,如果是就跳过下面的步骤
if n_id in closed_set:
continue
# 判断邻接节点是否是可行的,或者有没有超过范围，如果不可行，那么就直接跳过下面的操作，直接进入下一个循环
if not self.verify_node(node):
continue
# 如果这个节点还不在open_set里面，那就把他加进去
if n_id not in open_set:
open_set[n_id] = node # Discover a new node
# 否则
else:
# 其实这一步的目的就是说我有了一个新的临近节点，而原本已经存在与open_set中的节点与这个新的邻接节点距离更近，
# 所以我们就需要更新一下对应的距离
if open_set[n_id].cost >= node.cost:
# This path is the best until now. record it!
open_set[n_id] = node
# 计算最后的路径
rx, ry = self.calc_final_path(goal_node, closed_set)
return rx, ry
def calc_final_path(self, goal_node, closed_set):
# generate final course
# 直接计算的就是xy的坐标值
rx, ry = [self.calc_position(goal_node.x, self.min_x)], [
self.calc_position(goal_node.y, self.min_y)]
parent_index = goal_node.parent_index
# 知道初始节点
while 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_index
return rx, ry
def calc_position(self, index, minp):
pos = index * self.resolution + minp
return pos
def calc_xy_index(self, position, minp):
return round((position - minp) / self.resolution)
def calc_index(self, node):
# 先算有多少个整行,再算剩余的一行
return node.y * self.x_width + node.x
def verify_node(self, node):
# 计算x，y坐标
px = self.calc_position(node.x, self.min_x)
py = self.calc_position(node.y, self.min_y)
# 判断是否在地图的范围里面
if px < self.min_x:
return False
if py < self.min_y:
return False
if px >= self.max_x:
return False
if py >= self.max_y:
return False
# 判断是否在障碍物里面
if self.obstacle_map[node.x][node.y]:
return False
# 如果上面的判断都不是，那么就说明该邻接节点是可以行走的
return True
def calc_obstacle_map(self, ox, oy):
''' 第1步：构建栅格地图 '''
# 第一步是获得地图的边界,四个角的位置
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)
# 计算X方向和Y方向栅格的个数
self.x_width = round((self.max_x - self.min_x) / self.resolution)
self.y_width = round((self.max_y - self.min_y) / self.resolution)
# 打印输出栅格的大小
print("x_width:", self.x_width)
print("y_width:", self.y_width)
# obstacle map generation
# 初始化地图,采用两层的列表来表示,初始化为False说明都还没有设置障碍物
self.obstacle_map = [[False for _ in range(self.y_width)]
for _ in range(self.x_width)]
# 遍历栅格
for ix in range(self.x_width):
# 计算具体的栅格x位置
x = self.calc_position(ix, self.min_x)
# 计算具体的栅格的y的位置
for iy in range(self.y_width):
y = self.calc_position(iy, self.min_y)
for iox, ioy in zip(ox, oy):
# 障碍物到栅格的距离
d = math.hypot(iox - x, ioy - y)
# 膨胀障碍物
if d <= self.robot_radius:
# 障碍物的栅格地图设置为True
self.obstacle_map[ix][iy] = True
break
@staticmethod
def get_motion_model():
# dx, dy具体的行走 方向,cost行走的代价
# 往x方向(右)走1
motion = [[1, 0, 1],
# 往y方向(上)走1
[0, 1, 1],
# 左
[-1, 0, 1],
# 下
[0, -1, 1],
# 斜下
[-1, -1, math.sqrt(2)],
# 斜上
[-1, 1, math.sqrt(2)],
[1, -1, math.sqrt(2)],
[1, 1, math.sqrt(2)]]
return motion
def main():
# start and goal position
sx = -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 positions
ox, 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 cover
plt.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 cover
plt.plot(rx, ry, "-r")
plt.pause(0.01)
plt.show()
if __name__ == '__main__':
main()
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