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of zero. Each time data needs to be put on the stack it should be stored at the position indicated by StackPtr. After the data is stored, the pointer is incremented so that it always points to the next available position in the array. When the last data stored in the array is needed, we retrieve it from position StackPtr1. When all the exits have failed, we discard the data for the cell by decrementing the value of StackPtr. Once the maze has been solved, the array Stack will contain only the data (the choices) that worked for each of the multi-exit cells in the correct path through the maze. If the robot tries to run the maze again, it simply has to pull the next piece of data from the Stack each time a cell with multiple exits is encountered. The data pulled will indicate which choices are left in the variable dir. The rst choice indicated is the last choice the robot made for this cell, and consequently must be the correct choice. The subroutine SolveMaze in Fig. 14.5 accomplishes all the above logic. The code is well documented and follows the algorithm described above. The subroutine CheckBefore in Fig. 14.5, determines if the current cell has been previously visited. This is accomplished by saving the robot s GPS coordinates the rst time a cell with multiple exits is visited. Later when the robot wishes to see if it has been here previously, it uses this subroutine to compare its current GPS coordinates with the ones saved. Since the robot may not be in exactly the same place as it was previously, the distance between the two points is calculated using the pythagorean theorem [using the PolarR() function]. If this distance is less than 10 pixels, we assume that the cell has been visited before. 14.3.5 RENEGOTIATING THE MAZE As mentioned earlier, once the robot solves the maze, the program uses the subroutine RetraceMaze to place the robot back at its original starting position so it can try again. On the second attempt, the robot uses the saved data to make the correct turn each time it encounters a cell with multiple exits. Also, on the second run through the maze, the robot uses its pen to highlight the path it takes. 14.3.6 EMBEDDED DEBUG COMMANDS If you examine the code in Fig. 14.5 you will see that there are numerous Debug commands inside the code. None of these commands will execute unless you remove the comment symbols from the rst line in the program. Do this and then execute the program. As the robot progresses through the maze the debug window will pop-up at appropriate situations and tell you what the robot is doing. When this happens analyze the displayed information and click Step to proceed. Hopefully you will nd this debugging exercise helpful in understanding the algorithm discussed in this section.
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14.4 A Mapped Solution
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When we decide to go from one place to another on the highways, we do not just get in the car and drive in the general direction of our destination taking roads that might lead there. Rather we consult a map and plan a route through the maze of roads and then follow that route.
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Our robot can be programmed to do the same thing if it is given a map of the maze. The robot can then consult the map to plan a route from the start position to the goal position. The robot would then follow the planned route. This way the robot can determine ahead of time if it is impossible to reach the goal from where it is and not waste time if it cannot. 14.4.1 MAPPING THE MAZE In order to achieve this algorithm a GPS and a map will be needed. How can we represent this map in the most ef cient way for the robot Since the computer generates the maze randomly, the computer must generate the map as well. Highway and road maps are represented on computers and GPS devices as a list of junctions and a connectivity list showing how the junctions are connected. In computer science this type of data structure is called a graph. There are various types of graphs that differ in how the connection information is represented. Directional graphs represent one-way connections. Weighted graphs represent the distance of connections. Simple graphs are just a connection list where a connection does not have a weight and is bidirectional. This is the type of graph we will use for our maze map. The array in Fig. 14.8 is a representation of the graph shown below it. A 1 in the array means that there is a connection. A 0 means no connection. Notice that the matrix is symmetrical, which means that if junction 1 is connected to junction 2 then there is also a connection from junction 2 to junction 1. If the graph is not bidirectional then it is not necessarily symmetrical. Also, notice that all the numbers are 1. If the graph is weighted then the numbers represent the length of the connections (like in roads). We will do the same and represent our maze as an array of connections. This will allow the algorithm to gure out how to go from one place to another using this connectivity array. The graph array will be generated automatically from the array Maze[ ]. The elements in each row of the array represent the nodes that have a direct connection to the node with the same number as the row. So, for instance, if you look at row 2 you know that from node 2 you can go directly to nodes 1 and 3, but not node 4. In this example there is no connection from the node to itself (in other scenarios this may not be the case). You can see that the array is a true representation of the diagram below it. In summary, if Graph[I,J] 1 we know there is a direct path from node I to node J, if it is 0 it means that there is no direct path.
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