原理
A*也是一种基于图的搜索算法，算法原理很简单，每次弹出一个代价最小的结点，是一个优先级队列。其实老实讲，A* = Dijkstra + Greedy Best First Search。
1）我们知道Dijkstra是一个最优的算法，也是一个优先级队列，其结点的弹出是基于从起点出发到其他结点的代价，它的缺点在于没有任何关于到终点的先验，有点类似于广度优先搜索；
2）同样的，Greedy Best First Search，也是一个优先级队列，其结点的弹出是基于结点到终点的代价，它的缺点在于不一定保证是最优的；
我们结合上述两点，推导出了A*算法，设计了一个新的代价函数f = g + h，其中g代表了从起点出发到结点的累计代价；而h则代表了结点到终点的代价。
以上基本粗糙的描述了A*算法，接下来我们看一下其他的资料！
其他资料
以上这个PPT如果按照A*算法，那么路径是S-G，但并不是最优的；最优的应该是S-A-G；那么哪里出现了问题呢，是h这个启发式函数的估计出现了问题！那么怎么解决这个问题呢，怎么保证A*是最优的呢？我们必须要保证我们的设计的H这个启发式函数一定是小于实际上结点到终点的代价！
欧式距离这个启发式函数肯定满足上述条件，所以欧时距离这个启发式函数一定保证A*是最优的；
曼哈顿距离不一定满足上述条件，如果机器人可以沿着对角线运动，那么不满足上述条件，如果不能沿着对角线运动，那么满足上述条件，所以曼哈顿距离这个启发式函数不一定保证A*是最优的。
0也保证了A*是最优的，因为启发式函数为0，A*退化为Dijkstra，肯定也是最优的。
Weighted A* 次优的，偏向于Greedy Best Frist Search，但速度可能更快！
我们之前找到了启发式函数远远小于实际上结点到终点的代价，因此呢搜索了很多无关的结点。
我们找到了一个最好的启发式函数！
打破平衡性，其实际上由于搜索到的很多结点具有相等的代价，因此扩展的时候没有倾向性，所以扩展了很多无关的结点。那么如何解决这个问题呢？参考如下方法：
1）令启发式函数稍微的放大，但稍微影响了最优的结果，不过基本无法影响最后的最优性；
2）如果f相同，那么比较启发式函数；
3）给每个格子预先设计一个很小的随机数或者是给启发式函数或者是给累计代价；
4）给搜索的路径一个倾向性；
例子
这个例子是在开源代码中找到的，给添加了一些注释；另外该算法虽然是使用python写的，但其实可以忽略掉一些辅助性的东西而关注于核心思想。
c_id = min(open_set, key=lambda o: open_set[o].cost + self.calc_heuristic(goal_node, open_set[o]))代价最小的结点弹出！
"""
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()
其栅格地图如下图所示：
其最终搜索的路径如下图所示：
我们看下加权A*的效果如下所示，启发式权重为2：
