路径规划与轨迹优化 —— Dijkstra算法寻找最短路径
一、算法思路
Dijkstra算法是一种用来寻找最短路径的算法，其中涉及的思想有贪心、动态规划、广度优先搜索等。
图中g(n)代表的时代价，在机器人路径规划中可以理解为距离。
二、代码
源码来源于Github ，我自己做了一些注解。算法过程不难，主要注意几个细节：
- 原始地图坐标和栅格地图坐标的转换
- 需要对障碍物处做膨胀处理（机器人半径制约了邻近障碍物的栅格）
- 为了标记节点，又加了一个一维数组的索引表示
- 每次找到一个节点时，都要记录上一个节点（通过上一条提到的索引），这样可以记录下完整的路径。
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
self.resolution = resolution
self.robot_radius = robot_radius
self.calc_obstacle_map(ox, oy) #构建栅格地图（包含障碍物的膨胀）
self.motion = self.get_motion_model() #规定路径权重
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 前一个结点
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
"""
#原始坐标转换成栅格坐标并赋给节点
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
#calc_index是对节点的一个遍历标号 方便对节点的操作和定位 相当于key
open_set[self.calc_index(start_node)] = start_node #起点先加入open_set
while 1:
#c_id取的是最小代价点的key
c_id = min(open_set, key=lambda o: open_set[o].cost) # 取cost最小的节点
#当前节点
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
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) #c_id传入的是上一个被收录点的遍历索引
n_id = self.calc_index(node)
if n_id in closed_set: #如果相邻点已经在closed_list 就不用收录了
continue
if not self.verify_node(node): #符合地图的约束
continue
if n_id not in open_set: #将新发现的节点收录
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 #更新cost
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.x_width + node.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):
''' 第1步：构建栅格地图 '''
#四个顶点
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)
#栅格个数
self.x_width = round((self.max_x - self.min_x) / self.resolution)
self.y_width = round((self.max_y - self.min_y) / self.resolution)
print("x_width:", self.x_width)
print("y_width:", self.y_width)
# obstacle map generation
# 初始化地图
self.obstacle_map = [[False for _ in range(self.y_width)]
for _ in range(self.x_width)]
# 设置障碍物
# x和y是栅格地图中的坐标 对障碍物做膨胀处理时要找到x和y在于原始地图的坐标
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) #原始地图中所有栅格距离障碍物的距离
if d <= self.robot_radius: #障碍物到附近点的距离小于机器人半径 机器人不能通过
self.obstacle_map[ix][iy] = True
break
@staticmethod
def get_motion_model(): #相当于赋予权重
# dx, dy, cost（x轴方向，y轴方向，代价（距离））
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():
# start and goal position
#起点
sx = -5.0 # [m]
sy = -5.0 # [m]
#终点
gx = 50.0 # [m]
gy = 50.0 # [m]
grid_size = 2.0 # [m] #栅格大小
robot_radius = 1.0 # [m] #机器人半径
# set obstacle positions
ox, oy = [], [] #ox oy中存放的是障碍物在原始地图的坐标值
#四周墙面
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)
#障碍物
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)
#对墙面和障碍物画图
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")
#创建Dijkstra对象
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()