Python实现Dijkstra算法与A*改进：机器人路径规划详解
发布时间：2026-10-15 |
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本文详细介绍如何使用Python实现Dijkstra算法进行机器人路径规划，涵盖算法原理、集合操作及代码实现。同时对比分析A*算法通过引入启发式函数优化搜索效率的方法，结合PythonRobotics项目源码，提供从理论到实践的完整路径规划解决方案。
前言
在机器人运动过程中，为了实现有效避障并获取最短路径，必须采用特定的路径规划算法。常见的路径规划方法包括Dijkstra算法和A*算法等。GitHub上的开源项目PythonRobotics提供了高质量的源代码与参考实现，通过研读这些代码，可以深入理解Dijkstra算法的核心机制与应用细节。
一、算法原理
如图所示，Dijkstra算法旨在解决有向加权图中的最短路径查找问题。图中红色节点1标识起始位置，蓝色节点6标识目标位置。连接节点的箭头上的数值代表两点间的距离，即在模型中定义的代价（cost）。
该算法依赖两个关键集合：open_set（开集）和closed_set（闭集），并遵循以下逻辑步骤执行：
- 将初始节点加入open_set中；
- 从open_set中选取代价最小的节点，将其移至closed_set并设为当前节点；
- 将与当前节点相邻的节点加入open_set，若计算出的新代价更小，则更新该节点的代价；
- 重复步骤2和3，直至找到终点节点。
需要注意的是，open_set中的节点代价是动态变化的，而closed_set中的节点代价已确定为最小值，这也是“open”与“close”命名的由来。
之所以closed_set中的代价是最小的，是因为算法采用了贪心策略。一旦节点被加入closed_set，意味着从初始点到该节点的距离已小于或等于到open_set中任何节点的距离，因此不可能再找到比当前更小的路径代价。
二、程序代码
"""
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 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_index):
self.x = x # index of grid
self.y = y # index of grid
self.cost = cost
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
"""
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_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 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_index = goal_node.parent_index
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):
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()
三、运行结果
四、 A*算法：Dijkstra算法的改进
Dijkstra算法本质上属于贪心搜索，其时间复杂度为O(n^2)。为了减少无效搜索次数，可以引入启发式函数（heuristic），例如计算当前节点到目标节点的距离。在选择open_set元素时，将评估函数调整为cost + heuristic，从而为搜索提供方向性指引，有效避免盲目搜索。通过运行PythonRobotics中的Astar代码，可获得如下优化结果：
总结
至此，关于使用Python编写Dijkstra算法进行机器人路径规划的详细介绍已完成。如需了解更多Python实现Dijkstra算法的相关内容，建议查阅本站往期文章或继续浏览后续系列教程，感谢大家的支持。
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