原理
dijkstra也是一種基於圖的搜索算法，算法原理也很簡單：按照代價排序每次彈出一個代價最小的結點，同樣也是一個優先級隊列，不過需要注意與Greedy Best First Search的區別（優先級隊列，按照與終點的距離排序，每次彈出距離最小的結點）。補充下，dijkstra一定是最優的！
根據累加代價最小的原則，dijkstra彈出節點的方式如下圖（1白色、2藍色、3綠色、4黃色）：
其他資料
例子
這個例子是在開源代碼中找到的，我給添加了一些註釋；另外該算法雖然是使用python寫的，但其實可以忽略掉一些輔助性的東西而關注於核心思想。
c_id = min(open_set, key=lambda o: open_set[o].cost) 相當於優先級隊列，彈出代價最小的結點。
"""
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 a star 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
self.resolution = resolution
self.robot_radius = robot_radius
self.calc_obstacle_map(ox, oy)
self.motion = self.get_motion_model()
class Node:
def __init__(self, x, y, cost, parent):
self.x = x # index of grid
self.y = y # index of grid
self.cost = cost
self.parent = parent # index of previous Node
def __str__(self):
return str(self.x) + "," + str(self.y) + "," + str(
self.cost) + "," + str(self.parent)
def planning(self, sx, sy, gx, gy):
"""
dijkstra path search
input:
sx: start x position [m]
sy: 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
"""
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_node
while 1:
c_id = min(open_set, key=lambda o: open_set[o].cost)
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)
if current.x == goal_node.x and current.y == goal_node.y:
print("Find goal")
goal_node.parent = current.parent
goal_node.cost = current.cost
break
# Remove the item from the open set
del open_set[c_id]
# Add it to the closed set
closed_set[c_id] = current
# expand search grid based on motion model
for 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:
continue
if not self.verify_node(node):
continue
if n_id not in open_set:
open_set[n_id] = node # Discover a new node
else:
if open_set[n_id].cost >= node.cost:
# This path is the best until now. record it!
open_set[n_id] = node
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
rx, ry = [self.calc_position(goal_node.x, self.min_x)], [
self.calc_position(goal_node.y, self.min_y)]
parent = goal_node.parent
while parent != -1:
n = closed_set[parent]
rx.append(self.calc_position(n.x, self.min_x))
ry.append(self.calc_position(n.y, self.min_y))
parent = n.parent
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):
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 False
if py < self.min_y:
return False
if px >= self.max_x:
return False
if py >= self.max_y:
return False
if self.obstacle_map[node.x][node.y]:
return False
return True
def 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 generation
self.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] = True
break
@staticmethod
def get_motion_model():
# dx, dy, cost
motion = [[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 motion
def main():
print(__file__ + " start!!")
# 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()
其柵格地圖如下圖所示：
其最終搜索的路徑如下圖所示：
