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FIGURE 14.8 The graph shown describes the maze below it.
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If you look at the diagram (or array), you can see that there is no direct path between 2 and 4. However it is still possible to go from 2 to 4 by following a path 2, 1, then 4 or 2, 3, then 4. In a much more complex graph it might not be so obvious if there is a path, and whether it is the shortest. Humans are amazing at being able to glance at diagrams like the mazes you saw in the previous sections (see Fig. 14.6 and 14.3) and guring out in seconds if there is a possible pathway between two points on the maze. We are also very adept at nding a path (not always the shortest) just by looking. A robot is not so fortunate. Using the above setup, computer routines can be developed to traverse the array of connections to determine a path, or even the optimal path. We will see how to do this in the routines we will develop in the following sections.
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14.4.2 THE PROGRAM The base program in Fig. 14.1 will be used to demonstrate how maps can improve the robot s navigational abilities. We will give only the new and modi ed subroutines in Fig. 14.9. The new main program section is not very different from before, except the messages are changed to re ect the fact that the robot now knows if it cannot reach the goal. The variable Found is set to true or false by the SolveMaze subroutine. All subroutines not listed in Fig. 14.9 are the same as in Fig. 14.1.
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14.4.3 CREATING THE MAP S GRAPH The routine BuildMapGraph creates the graph of the maze as discussed above. The array MapGraph[ ] will hold a 35 35 grid of numbers allowing it to represent a maze with up to 35 junctions. Each junction in the maze is given a number (0 34). The array will have the value 0 at the position [i, j] if there is no connection between junctions i and j. If there is a connection then the value will be 1. As you can see, since the maze is a two-way maze the position, [i, j] is the same as [j, i] (i.e., the matrix is symmetrical). The way to check if a junction is connected to another is by using the values stored in the Maze[ ] array which was created by the PlotMaze subroutine. If-statements are used to determine if a line radiates from the junction (north, south, east, and/or west) and if there are lines emanating from any adjacent junction at the same time. If there are, then the junctions are connected, otherwise they are not.
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14.4.4 SOLVING THE MAZE The routine SolveMaze calculates the junction nearest to the goal (GoalNode) and the junction nearest to the robot (RobotNode) using calls to the subroutine CalcNodeNumber. The procedure for calculating the nearest node takes into account that there must be a road between the goal or robot and that nearest node. BuildMapGraph is called when the program is run for the rst time. The routine then calls the FindPath subroutine (discussed below) which creates a path (if one is possible) in the array Stack[ ] starting at the RobotNode and ending at the GoalNode showing what intermediate nodes to take to create the path.
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MainProgram: FirstTime = true gosub DisplayInstructions while true gosub PlotMaze gosub PlaceRobot gosub SelectGoal gosub SolveMaze FirstTime = false if Found Message = "Goal Is Found" else Message = "Goal Is Unreachable" endif Message = Message+"---Press Any-Key or Right-Mouse" Message = Message + "-Button to repeat with same maze" gosub DisplayMessage Beep gosub WaitForKeyOrMouse wend End //============================================================= BuildMapGraph: Dim MapGraph[35,35] MConstant MapGraph,0 for i = 0 to 4 for j = 0 to 6 S = Maze[i,j] NN = i*7+j if (S&1) and (j<6) if Maze[i,j+1]&4 MapGraph[NN,NN+1] = 1 endif endif if (S&2) and (i>0) if Maze[i-1,j]&8 MapGraph[NN,NN-7] = 1 endif endif if (S&4) and (j>0) if Maze[i,j-1]&1 MapGraph[NN,NN-1] = 1 endif endif if (S&8) and (i<4) if Maze[i+1,j]&2 MapGraph[NN,NN+7] = 1 endif endif next next Return //============================================================= SolveMaze: //get Node Nearest to Goal Tx = Gx Ty = Gy gosub CalcNodeNumber GoalNode = NodeNumber
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