寫在前面的話：只是對兩種路徑優化算法進行簡單的理解和嘗試，爲後續使用做準備。如果用到，請再次好好理解原理和Matlab源碼。
參考博客：
直接上乾貨（參考上述博客可得）首先給出Matlab下的三個腳本文件：
TestScript.m
%
% TestScript for Assignment 1
%
%% Define a small map
% map = false(10);
map = ans;
% Add an obstacle
% map (1:5, 6) = true;
map = logical(map);
start_coords = [1, 1];
dest_coords = [40, 20];
%%
close all;
% [route, numExpanded] = DijkstraGrid (map, start_coords, dest_coords);
% Uncomment following line to run Astar
[route, numExpanded] = AStarGrid (map, start_coords, dest_coords);
%HINT: With default start and destination coordinates defined above, numExpanded for Dijkstras should be 76, numExpanded for Astar should be 23.
AStarGrid.m
function [route,numExpanded] = AStarGrid (input_map, start_coords, dest_coords)
% 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用一個map矩陣來表示每個點的狀態
% 1 - white - clear cell
% 2 - black - obstacle
% 3 - red = visited 相當於CLOSED列表的作用
% 4 - blue - on list 相當於OPEN列表的作用
% 5 - green - start
% 6 - yellow - destination
cmap = [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
% iteration
drawMapEveryTime = true;
[nrows, ncols] = size(input_map);
% map - a table that keeps track of the state of each grid cell用來上色的
map = zeros(nrows,ncols);
map(~input_map) = 1; % Mark free cells
map(input_map) = 2; % Mark obstacle cells
% Generate linear indices of start and dest nodes將下標轉換爲線性的索引值
start_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 information
parent = zeros(nrows,ncols);%用來記錄每個節點的父節點
%
[X, Y] = meshgrid (1:ncols, 1:nrows);
xd = dest_coords(1);
yd = dest_coords(2);
% Evaluate Heuristic function, H, for each grid cell
% Manhattan distance用曼哈頓距離作爲啓發式函數
H = abs(X - xd) + abs(Y - yd);
H = H';
% Initialize cost arrays
f = 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 expanded
numExpanded = 0;
% Main Loop
while 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,其中的current是index值，需要轉換
[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
numExpanded=numExpanded+1;
% 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
%
action=[-1 0; 1 0; 0 -1; 0 1];%上，下，左，右
for a=1:4
expand=[i,j]+action(a,:);
expand1=expand(1,1);
expand2=expand(1,2);
%不超出邊界，不穿越障礙，不在CLOSED列表裏，也不是起點，則進行擴展
if ( expand1>=1 && expand1<=nrows && expand2>=1 && expand2<=nrows && map(expand1,expand2)~=2 && map(expand1,expand2)~=3 && map(expand1,expand2)~=5)
if ( g(expand1,expand2)> g(i,j)+1 )
g(expand1,expand2)= g(i,j)+1;
f(expand1,expand2)= g(expand1,expand2)+H(expand1,expand2);
parent(expand1,expand2)=current;
map(expand1,expand2)=4;
end
end
end
%*********************************************************************
end
%% Construct route from start to dest by following the parent links
if (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;
end
end
end
DijkstraGrid.m
function [route,numExpanded] = DijkstraGrid (input_map, start_coords, dest_coords)
% Run Dijkstra's 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 - destination
cmap = [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
% iteration
drawMapEveryTime = true;
[nrows, ncols] = size(input_map);
% map - a table that keeps track of the state of each grid cell
map = zeros(nrows,ncols);
map(~input_map) = 1; % Mark free cells
map(input_map) = 2; % Mark obstacle cells
% Generate linear indices of start and dest nodes
start_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;
% Initialize distance array
distanceFromStart = Inf(nrows,ncols);
% For each grid cell this array holds the index of its parent
parent = zeros(nrows,ncols);
distanceFromStart(start_node) = 0;
% keep track of number of nodes expanded
numExpanded = 0;
% Main Loop
while 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 distance
[min_dist, current] = min(distanceFromStart(:));
if ((current == dest_node) || isinf(min_dist))
break;
end;
% Update map
map(current) = 3; % mark current node as visited
numExpanded=numExpanded+1;
% Compute row, column coordinates of current node
[i, j] = ind2sub(size(distanceFromStart), current);
% *********************************************************************
% YOUR CODE BETWEEN THESE LINES OF STARS
% Visit each neighbor of the current node and update the map, distances
% and parent tables appropriately.
action=[-1 0; 1 0; 0 -1; 0 1];%上，下，左，右
for a=1:4
expand=[i,j]+action(a,:);
expand1=expand(1,1);
expand2=expand(1,2);
%不超出邊界，不穿越障礙，不在CLOSED列表裏，則進行擴展
if ( expand1>=1 && expand1<=nrows && expand2>=1 && expand2<=ncols && map(expand1,expand2)~=2 && map(expand1,expand2)~=3 && map(expand1,expand2)~=5 )
% if ( expand1>=1 && expand1<=nrows && expand2>=1 && expand2<=ncols && map(expand1,expand2)~=2 && map(expand1,expand2)~=3 && map(expand1,expand2)~=5)
if ( distanceFromStart(expand1,expand2)> distanceFromStart(i,j)+1 )
distanceFromStart(expand1,expand2)= distanceFromStart(i,j)+1;
parent(expand1,expand2)=current;
map(expand1,expand2)=4;
end
end
end
distanceFromStart(current) = Inf; % remove this node from further consideration
%*********************************************************************
end
%% Construct route from start to dest by following the parent links
if (isinf(distanceFromStart(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;
end
end
end
注： 運行環境，Matlab 2019a 版本，安裝 RTB（Robotic Tool Box） 工具包，鏈接地址爲，RTB安裝鏈接 。該工具包中可以運行作者大佬寫到的matlab/simulink四軸歷程，只需要使用指令 sl_quadrotor 即可。
使用方法：
在Matlab 中，使用 makemap(30) 來生成地圖，通過鼠標來設置障礙形狀。該例子生成了一個30*30的方陣，然後直接運行TestScript.m即可。其中要在TestScript.m中選擇是採用A算法，還是Dijkstra算法。同時設置起始點和終點在哪。下圖顯示得到的A算法路徑優化結果。其中綠色點爲起點，黃色點爲終點，黑色表示障礙，白色表示空閒，紅色表示搜尋過，灰色表示最後規劃的路徑。
下圖顯示Dijkstra算法的路徑優化結果：
對應的動態效果已經錄屏，下面給出傳送門（錄屏水印廣告請忽略）：
通過對比可以看出：A* 算法搜索速度較快，畢竟裏面有貪心算法。這在地圖較大的場景應用較好。但是A*算法只能得到局部最優解，並不能保證全局最優解。相比之下，Dijkstra算法儘管搜索速度慢，但是是全局最優解。不知道兩種方法結合gmapping，hector或者cartographer生成的柵格地圖會是什麼樣的效果。後面期待嘗試一下。