原理
A*也是一種基於圖的搜索算法，算法原理很簡單，每次彈出一個代價最小的結點，是一個優先級隊列。其實老實講，A* = Dijkstra + Greedy Best First Search。
1）我們知道Dijkstra是一個最優的算法，也是一個優先級隊列，其結點的彈出是基於從起點出發到其他結點的代價，它的缺點在於沒有任何關於到終點的先驗，有點類似於廣度優先搜索；
2）同樣的，Greedy Best First Search，也是一個優先級隊列，其結點的彈出是基於結點到終點的代價，它的缺點在於不一定保證是最優的；
我們結合上述兩點，推導出了A*算法，設計了一個新的代價函數f = g + h，其中g代表了從起點出發到結點的累計代價；而h則代表了結點到終點的代價。
以上基本粗糙的描述了A*算法，接下來我們看一下其他的資料！
其他資料
以上這個PPT如果按照A*算法，那麼路徑是S-G，但並不是最優的；最優的應該是S-A-G；那麼哪裏出現了問題呢，是h這個啓發式函數的估計出現了問題！那麼怎麼解決這個問題呢，怎麼保證A*是最優的呢？我們必須要保證我們的設計的H這個啓發式函數一定是小於實際上結點到終點的代價！
歐式距離這個啓發式函數肯定滿足上述條件，所以歐時距離這個啓發式函數一定保證A*是最優的；
曼哈頓距離不一定滿足上述條件，如果機器人可以沿着對角線運動，那麼不滿足上述條件，如果不能沿着對角線運動，那麼滿足上述條件，所以曼哈頓距離這個啓發式函數不一定保證A*是最優的。
0也保證了A*是最優的，因爲啓發式函數爲0，A*退化爲Dijkstra，肯定也是最優的。
Weighted A* 次優的，偏向於Greedy Best Frist Search，但速度可能更快！
我們之前找到了啓發式函數遠遠小於實際上結點到終點的代價，因此呢搜索了很多無關的結點。
我們找到了一個最好的啓發式函數！
打破平衡性，其實際上由於搜索到的很多結點具有相等的代價，因此擴展的時候沒有傾向性，所以擴展了很多無關的結點。那麼如何解決這個問題呢？參考如下方法：
1）令啓發式函數稍微的放大，但稍微影響了最優的結果，不過基本無法影響最後的最優性；
2）如果f相同，那麼比較啓發式函數；
3）給每個格子預先設計一個很小的隨機數或者是給啓發式函數或者是給累計代價；
4）給搜索的路徑一個傾向性；
例子
這個例子是在開源代碼中找到的，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 + self.calc_heuristic(goal_node, open_set[o]))代價最小的結點彈出！
"""
A* grid planning
author: Atsushi Sakai(@Atsushi_twi)
Nikos Kanargias ([email protected])
See Wikipedia article (https://en.wikipedia.org/wiki/A*_search_algorithm)
"""
import math
import matplotlib.pyplot as plt
show_animation = True
class AStarPlanner:
def __init__(self, ox, oy, resolution, rr):
"""
Initialize grid 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.resolution = resolution
self.rr = rr
self.min_x, self.min_y = 0, 0
self.max_x, self.max_y = 0, 0
self.obstacle_map = None
self.x_width, self.y_width = 0, 0
self.motion = self.get_motion_model()
self.calc_obstacle_map(ox, oy)
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
self.parent_index = parent_index
def __str__(self):
return str(self.x) + "," + str(self.y) + "," + str(
self.cost) + "," + str(self.parent_index)
def planning(self, sx, sy, gx, gy):
"""
A star path search
input:
s_x: start x position [m]
s_y: start y position [m]
gx: goal x position [m]
gy: goal y 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_grid_index(start_node)] = start_node
while 1:
if len(open_set) == 0:
print("Open set is empty..")
break
c_id = min(
open_set,
key=lambda o: open_set[o].cost + self.calc_heuristic(goal_node, open_set[o]))
current = open_set[c_id]
# show graph
if show_animation: # pragma: no cover
plt.plot(self.calc_grid_position(current.x, self.min_x),
self.calc_grid_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_index
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_grid search grid based on motion model
for i, _ in enumerate(self.motion):
node = self.Node(current.x + self.motion[i][0],
current.y + self.motion[i][1],
current.cost + self.motion[i][2], c_id)
n_id = self.calc_grid_index(node)
# If the node is not safe, do nothing
if not self.verify_node(node):
continue
if n_id in closed_set:
continue
if n_id not in open_set:
open_set[n_id] = node # discovered 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_grid_position(goal_node.x, self.min_x)], [self.calc_grid_position(goal_node.y, self.min_y)]
parent_index = goal_node.parent_index
while parent_index != -1:
n = closed_set[parent_index]
rx.append(self.calc_grid_position(n.x, self.min_x))
ry.append(self.calc_grid_position(n.y, self.min_y))
parent_index = n.parent_index
return rx, ry
@staticmethod
def calc_heuristic(n1, n2):
w = 1.0 # weight of heuristic
d = w * math.hypot(n1.x - n2.x, n1.y - n2.y)
return d
def calc_grid_position(self, index, min_position):
"""
calc grid position
:param index:
:param min_position:
:return:
"""
pos = index * self.resolution + min_position
return pos
def calc_xy_index(self, position, min_pos):
return round((position - min_pos) / self.resolution)
def calc_grid_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_grid_position(node.x, self.min_x)
py = self.calc_grid_position(node.y, self.min_y)
if px < self.min_x:
return False
elif py < self.min_y:
return False
elif px >= self.max_x:
return False
elif py >= self.max_y:
return False
# collision check
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_grid_position(ix, self.min_x)
for iy in range(self.y_width):
y = self.calc_grid_position(iy, self.min_y)
for iox, ioy in zip(ox, oy):
d = math.hypot(iox - x, ioy - y)
if d <= self.rr:
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 = 10.0 # [m]
sy = 10.0 # [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")
a_star = AStarPlanner(ox, oy, grid_size, robot_radius)
rx, ry = a_star.planning(sx, sy, gx, gy)
if show_animation: # pragma: no cover
plt.plot(rx, ry, "-r")
plt.pause(0.001)
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：
我們看下加權A*的效果如下所示，啓發式權重爲2：
