路径规划A*算法

A*算法是在Dijkstra算法上进行改进，毕竟，我们是知道终点和起点的位置信息的，Dijkstra算法完全是四面八方全都找，然而我们既然已经知道，比方说重点在起点的北方，那么完全可以直接往北方搜索。

所以算法综合了Best-First Search和Dijkstra算法的优点：在进行启发式搜索提高算法效率的同时，可以保证找到一条最优路径（基于评估函数）。

在此算法中，如果以g(n)表示从起点到任意顶点n的实际距离，h(n)表示任意顶点n到目标顶点的估算距离（根据所采用的评估函数的不同而变化），那么A*算法的估算函数为：

f(x)=g(x)+h(x)

• 如果g(n)为0，即只计算任意顶点 n到目标的评估函数 h(n)，而不计算起点到顶点n的距离，则算法转化为使用贪心策略的Best-First Search，速度最快，但可能得不出最优解；
• 如果h(n)不高于实际到目标顶点的距离，则一定可以求出最优解，而且h(n)越小，需要计算的节点越多，算法效率越低，常见的评估函数有——欧几里得距离、曼哈顿距离、切比雪夫距离；

• 如果h(n)为0，即只需求出起点到任意顶点n的最短路径 g(n)，而不计算任何评估函数h(n)，则转化为单源最短路径问题，即Dijkstra算法，此时需要计算最多的定点；

• 4方向用L1距离

• 8方向用L∞距离，也就是max
• 任意方向用L2距离

heuristic function

这个评估函数，其实就是给搜索算法一个知识，告诉搜索算法往哪个方向比较对。
评估函数有这么几个特性：

• 将每一个节点映射到非负实数H:node→{x|x≥0,x∈R}
• H(goal)=0
• 对任意两个相邻的节点x,y
  
  • H(x)≤H(y)+d(x,y)
  • d(x,y)=weightheight of edge from x to y

这些属性可以令：

∀n,H(n)≤ shortest path from n to goal

关于H函数：
http://theory.stanford.edu/~amitp/GameProgramming/Heuristics.html

伪代码

• 对于每一个node n
  
  • n.f = Infinity, n.g = Infinity
• 创建一个空的List
• start.g = 0, start.f = H(start)，将start加入到list里边
• While(List非空)
  
  • current = list中最小f值的那个node
  • 如果current == goal，那么就完成啦
  • 对每一个node， n是这个node的临近的节点
  • if(n.g > (current.g + cost of edge from n to current))
    
    • n.g = current.g + cost of edge from n to current
    • n.f = n.g + H(n)
    • n.parent = current
    • add n to list if it’s not there already

matlab代码

function [route,numExpanded] = AStarGrid (input_map, start_coords, dest_coords, drawMapEveryTime)% Run A* algorithm on a grid.% Inputs : %   input_map : a logical array where the freespace cells are false or 0 and%   the obstacles are true or 1%   start_coords and dest_coords : Coordinates of the start and end cell%   respectively, the first entry is the row and the second the column.% Output :%    route : An array containing the linear indices of the cells along the%    shortest route from start to dest or an empty array if there is no%    route. This is a single dimensional vector%    numExpanded: Remember to also return the total number of nodes%    expanded during your search. Do not count the goal node as an expanded node. % set up color map for display% 1 - white - clear cell% 2 - black - obstacle% 3 - red = visited% 4 - blue  - on list% 5 - green - start% 6 - yellow - destinationcmap = [1 1 1; ...    0 0 0; ...    1 0 0; ...    0 0 1; ...    0 1 0; ...    1 1 0; ...    0.5 0.5 0.5];colormap(cmap);% variable to control if the map is being visualized on every% iterationdrawMapEveryTime = true;[nrows, ncols] = size(input_map);% map - a table that keeps track of the state of each grid cellmap = zeros(nrows,ncols);map(~input_map) = 1;   % Mark free cellsmap(input_map)  = 2;   % Mark obstacle cells% Generate linear indices of start and dest nodesstart_node = sub2ind(size(map), start_coords(1), start_coords(2));dest_node  = sub2ind(size(map), dest_coords(1),  dest_coords(2));map(start_node) = 5;map(dest_node)  = 6;% meshgrid will `replicate grid vectors' nrows and ncols to produce% a full grid% type `help meshgrid' in the Matlab command prompt for more informationparent = zeros(nrows,ncols);% [X, Y] = meshgrid (1:ncols, 1:nrows);xd = dest_coords(1);yd = dest_coords(2);% Evaluate Heuristic function, H, for each grid cell% Manhattan distanceH = abs(X - xd) + abs(Y - yd);H = H';% Initialize cost arraysf = Inf(nrows,ncols);g = Inf(nrows,ncols);g(start_node) = 0;f(start_node) = H(start_node);% keep track of the number of nodes that are expandednumExpanded = 0;% Main Loopwhile true    % Draw current map    map(start_node) = 5;    map(dest_node) = 6;    % make drawMapEveryTime = true if you want to see how the     % nodes are expanded on the grid.     if (drawMapEveryTime)        image(1.5, 1.5, map);        grid on;        axis image;        drawnow;    end    % Find the node with the minimum f value    [min_f, current] = min(f(:));    if ((current == dest_node) || isinf(min_f))        break;    end;    % Update input_map    map(current) = 3;    f(current) = Inf; % remove this node from further consideration    % Compute row, column coordinates of current node    [i, j] = ind2sub(size(f), current);    % *********************************************************************    % ALL YOUR CODE BETWEEN THESE LINES OF STARS    % Visit all of the neighbors around the current node and update the    % entries in the map, f, g and parent arrays    %    numExpanded = numExpanded + 1;    if(i-1>=1) %upper        id = sub2ind(size(map), i-1, j);            if((map(id) ~= 2) ... %if not obst            && (map(id) ~= 3) ... % if not visited            && (map(id) ~= 5)) ... % if not start            if(g(id) >= g(current) + 1)                g(id) = g(current) + 1;                f(id) = g(id) + H(id);                parent(id) = current;                map(id) = 4;            end                    end    end    if(i+1 <= nrows) %lower        id = sub2ind(size(map), i+1, j);        if((map(id) ~= 2) ... %if not obst            && (map(id) ~= 3) ... % if not visited            && (map(id) ~= 5)) ... % if not start            if(g(id) >= g(current) + 1)                g(id) = g(current) + 1;                f(id) = g(id) + H(id);                parent(id) = current;                map(id) = 4;            end                             end    end    if(j-1 >= 1) %left        id = sub2ind(size(map), i, j-1);        if((map(id) ~= 2) ... %if not obst            && (map(id) ~= 3) ... % if not visited            && (map(id) ~= 5)) ... % if not start            if(g(id) >= g(current) + 1)                g(id) = g(current) + 1;                f(id) = g(id) + H(id);                parent(id) = current;                map(id) = 4;            end                            end    end    if(j+1 <= ncols) %left        id = sub2ind(size(map), i, j+1);        if((map(id) ~= 2) ... %if not obst            && (map(id) ~= 3) ... % if not visited            && (map(id) ~= 5)) ... % if not start            if(g(id) >= g(current) + 1)                g(id) = g(current) + 1;                f(id) = g(id) + H(id);                parent(id) = current;                map(id) = 4;            end                           end    end    %*********************************************************************end%% Construct route from start to dest by following the parent linksif (isinf(f(dest_node)))    route = [];else    route = [dest_node];    while (parent(route(1)) ~= 0)        route = [parent(route(1)), route];    end    % Snippet of code used to visualize the map and the path    for k = 2:length(route) - 1                map(route(k)) = 7;        pause(0.1);        image(1.5, 1.5, map);        grid on;        axis image;    endendend

测试代码

map = false(10); %Input map parametersmap (2:10, 6) = true; %Obstacle Declarationstart_coords = [6, 2]; %Starting coordinatesdest_coords  = [8, 9]; %Destination CoordinatesdrawMapEveryTime = false; %Display Outputs[route, numExpanded] = AStarGrid(map, start_coords, dest_coords,drawMapEveryTime) %Implementatio

结果：