机器人路径规划算法是机器人领域的重要研究内容，用于在给定环境中为机器人找到一条从起始点到目标点的最优或可行路径。以下是几种常见的机器人路径规划算法及其细节：
### A*算法
A*算法是一种启发式搜索算法，结合了Dijkstra算法的最优路径搜索特性和贪心最佳优先搜索算法的高效性。它通过维护一个开放列表和一个关闭列表，不断扩展节点直到找到目标节点。其核心公式为 $f(n) = g(n) + h(n)$，其中 $g(n)$ 是从起始节点到当前节点的实际代价，$h(n)$ 是从当前节点到目标节点的启发式估计代价。
```python
import heapq
def heuristic(a, b):
# 曼哈顿距离作为启发式函数
return abs(a[0] - b[0]) + abs(a[1] - b[1])
def astar(array, start, goal):
neighbors = [(0, 1), (0, -1), (1, 0), (-1, 0)]
close_set = set()
came_from = {}
gscore = {start: 0}
fscore = {start: heuristic(start, goal)}
open_heap = []
heapq.heappush(open_heap, (fscore[start], start))
while open_heap:
current = heapq.heappop(open_heap)[1]
if current == goal:
data = []
while current in came_from:
data.append(current)
current = came_from[current]
return data
close_set.add(current)
for i, j in neighbors:
neighbor = current[0] + i, current[1] + j
tentative_g_score = gscore[current] + heuristic(current, neighbor)
if 0 <= neighbor[0] < array.shape[0]:
if 0 <= neighbor[1] < array.shape[1]:
if array[neighbor[0]][neighbor[1]] == 1:
continue
else:
# 越界
continue
else:
# 越界
continue
if neighbor in close_set and tentative_g_score >= gscore.get(neighbor, 0):
continue
if tentative_g_score < gscore.get(neighbor, 0) or neighbor not in [i[1] for i in open_heap]:
came_from[neighbor] = current
gscore[neighbor] = tentative_g_score
fscore[neighbor] = tentative_g_score + heuristic(neighbor, goal)
heapq.heappush(open_heap, (fscore[neighbor], neighbor))
return None
```
### Dijkstra算法
Dijkstra算法是一种广度优先搜索算法，用于在加权图中找到从单个源节点到所有其他节点的最短路径。它通过不断扩展距离源节点最近的节点，更新其邻居节点的距离，直到所有可达节点都被访问。
```python
import heapq
def dijkstra(graph, start):
distances = {node: float('inf') for node in graph}
distances[start] = 0
priority_queue = [(0, start)]
while priority_queue:
current_distance, current_node = heapq.heappop(priority_queue)
if current_distance > distances[current_node]:
continue
for neighbor, weight in graph[current_node].items():
distance = current_distance + weight
if distance < distances[neighbor]:
distances[neighbor] = distance
heapq.heappush(priority_queue, (distance, neighbor))
return distances
```
### 人工势场法
人工势场法将机器人在环境中的运动看作是在虚拟力场中的运动。目标点对机器人产生引力，障碍物对机器人产生斥力，机器人在合力的作用下向目标点移动。该方法计算简单，实时性好，但容易陷入局部极小值。
### 遗传算法
遗传算法是一种基于生物进化原理的优化算法。它通过模拟自然选择和遗传机制，对路径进行编码，生成初始种群，然后通过选择、交叉和变异等操作不断进化种群，直到找到最优路径。
### RRT（快速随机树）算法
RRT算法是一种基于采样的路径规划算法，通过随机采样在状态空间中快速扩展一棵树，直到树的某个节点接近目标点。该算法适用于高维空间和复杂环境，但生成的路径通常不是最优的。
```python
import numpy as np
import matplotlib.pyplot as plt
class RRT:
def __init__(self, start, goal, obstacle_list, rand_area, expand_dis=3.0, path_resolution=0.5, goal_sample_rate=5, max_iter=500):
self.start = Node(start[0], start[1])
self.goal = Node(goal[0], goal[1])
self.obstacle_list = obstacle_list
self.min_rand = rand_area[0]
self.max_rand = rand_area[1]
self.expand_dis = expand_dis
self.path_resolution = path_resolution
self.goal_sample_rate = goal_sample_rate
self.max_iter = max_iter
self.node_list = []
def planning(self, animation=True):
self.node_list = [self.start]
for i in range(self.max_iter):
rnd_node = self.get_random_node()
nearest_node = self.get_nearest_node(self.node_list, rnd_node)
new_node = self.steer(nearest_node, rnd_node, self.expand_dis)
if self.check_collision(new_node, self.obstacle_list):
self.node_list.append(new_node)
if animation and i % 5 == 0:
self.draw_graph(rnd_node)
if self.calc_dist_to_goal(self.node_list[-1].x, self.node_list[-1].y) <= self.expand_dis:
final_node = self.steer(self.node_list[-1], self.goal, self.expand_dis)
if self.check_collision(final_node, self.obstacle_list):
return self.generate_final_course(len(self.node_list) - 1)
return None
def get_random_node(self):
if np.random.randint(0, 100) > self.goal_sample_rate:
rnd = Node(np.random.uniform(self.min_rand, self.max_rand),
np.random.uniform(self.min_rand, self.max_rand))
else: # goal point sampling
rnd = Node(self.goal.x, self.goal.y)
return rnd
def get_nearest_node(self, node_list, rnd_node):
dlist = [(node.x - rnd_node.x) ** 2 + (node.y - rnd_node.y) ** 2 for node in node_list]
minind = dlist.index(min(dlist))
return node_list[minind]
def steer(self, from_node, to_node, extend_length=float("inf")):
new_node = Node(from_node.x, from_node.y)
d, theta = self.calc_distance_and_angle(new_node, to_node)
new_node.path_x = [new_node.x]
new_node.path_y = [new_node.y]
if extend_length > d:
new_node.x = to_node.x
new_node.y = to_node.y
else:
new_node.x = from_node.x + extend_length * np.cos(theta)
new_node.y = from_node.y + extend_length * np.sin(theta)
n_expand = int(np.floor(extend_length / self.path_resolution))
for _ in range(n_expand):
new_node.path_x.append(new_node.path_x[-1] + self.path_resolution * np.cos(theta))
new_node.path_y.append(new_node.path_y[-1] + self.path_resolution * np.sin(theta))
new_node.path_x.append(new_node.x)
new_node.path_y.append(new_node.y)
new_node.parent = from_node
return new_node
def check_collision(self, node, obstacle_list):
for (ox, oy, size) in obstacle_list:
dx_list = [ox - x for x in node.path_x]
dy_list = [oy - y for y in node.path_y]
d_list = [dx * dx + dy * dy for (dx, dy) in zip(dx_list, dy_list)]
if min(d_list) <= size ** 2:
return False # collision
return True # safe
def calc_dist_to_goal(self, x, y):
dx = x - self.goal.x
dy = y - self.goal.y
return np.sqrt(dx ** 2 + dy ** 2)
def generate_final_course(self, goal_ind):
path = [[self.goal.x, self.goal.y]]
node = self.node_list[goal_ind]
while node.parent is not None:
path.append([node.x, node.y])
node = node.parent
path.append([node.x, node.y])
return path
def draw_graph(self, rnd=None):
plt.clf()
# 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 rnd is not None:
plt.plot(rnd.x, rnd.y, "^k")
for node in self.node_list:
if node.parent is not None:
plt.plot(node.path_x, node.path_y, "-g")
for (ox, oy, size) in self.obstacle_list:
self.plot_circle(ox, oy, size)
plt.plot(self.start.x, self.start.y, "xr")
plt.plot(self.goal.x, self.goal.y, "xr")
plt.axis("equal")
plt.axis([-2, 15, -2, 15])
plt.grid(True)
plt.pause(0.01)
@staticmethod
def plot_circle(x, y, size, color="-b"): # pragma: no cover
deg = list(range(0, 360, 5))
deg.append(0)
xl = [x + size * np.cos(np.deg2rad(d)) for d in deg]
yl = [y + size * np.sin(np.deg2rad(d)) for d in deg]
plt.plot(xl, yl, color)
@staticmethod
def calc_distance_and_angle(from_node, to_node):
dx = to_node.x - from_node.x
dy = to_node.y - from_node.y
d = np.sqrt(dx ** 2 + dy ** 2)
theta = np.arctan2(dy, dx)
return d, theta
class Node:
def __init__(self, x, y):
self.x = x
self.y = y
self.path_x = []
self.path_y = []
self.parent = None
```
### 相关问题
1. 这些路径规划算法在不同环境下的性能如何比较？
2. 如何改进A*算法以避免陷入局部最优解？
3. 遗传算法在路径规划中的收敛速度如何提高？
4. RRT算法生成的路径如何进行优化？
5. 人工势场法的局部极小值问题有哪些解决方法？
加权树路径权重相等问题求解
博客围绕加权树展开，给定带根节点且各节点有权重的非空树，需找出权重和等于给定数的所有路径。给出了输入规格说明，还展示了部分代码，如定义节点结构体、比较函数及深度优先搜索函数等。
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