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Exploring a Variant of A* Search Algorithm for Robotic Path Planning in Unknown Environments

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This project is intended as an exploration of a variant of the A* search algorithm covered in class, in the traditional application of Robotic Path Planning. In particular, we are going to look at a situation where the environment is not fully known in advance, and must be learned by the agent as it moves through it. A gridworld is a discretization of terrain into square cells that are either unblocked (traversable) or blocked. Consider the following problem an agent in a gridworld has to move from its current cell (S) to a goal cell, the location of a stationary target. The layout of the gridworld (what cells are blocked or unblocked) is unknown. These kinds of challenges arise frequently in robotics, where a mobile platform equipped with sensors builds a map of the world as it traverses an unknown environment.

Assume that the initial cell of the agent is unblocked. The agent can move from its current cell in the four main compass directions (east, south, west, north) to any adjacent cell, as long as that cell is unblocked and still part of the gridworld. All moves take one timestep for the agent, and thus have cost 1. (We can consider this in terms of energy consumption for the agent, for instance.) The agent always knows which (unblocked) cell it is currently in, and which (unblocked) cell the target is in. The agent knows that blocked cells remain blocked and unblocked cells remain unblocked but does not know initially which cells are blocked. However, it can observe or sense the status of some of its surrounding cells (corresponding to its field of view) and remember this information for future use. By exploring the maze, collecting information on blockages, and integrating that into the whole, the agent must reach the target as efficiently as possible. We may structure a basic agent in the following way the agent assumes that all cells are unblocked, until it observes them to be blocked, and all path planning is done with the current state of knowledge of the gridworld under this freespace assumption. In other words, it moves according to the following strategy:

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Ã¯Â¿Â½ Based on its current knowledge of the environment, it plans a path from its current position to the goal.

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Ã¯Â¿Â½ This path should be the shortest (presumed) unblocked path available.

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Ã¯Â¿Â½ The agent attempts to follow this path plan, observing cells in its field of view as it moves.

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Ã¯Â¿Â½ Observed blocked cells or unblocked cells are remembered, updating the agentÃ¯Â¿Â½s knowledge of the environment.

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Ã¯Â¿Â½ If the agent discovers a block in its planned path, it re-plans, based on its current knowledge of the environment.

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Ã¯Â¿Â½ The cycle repeats until the agent either a) reaches the target or b) determines that there is no unblocked path to the target.

1: Why does re-planning only occur when blocks are discovered on the current path? Why not whenever knowledge of the environment is updated?

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2: Will the agent ever get stuck in a solvable maze? Why or why not?

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3: Once the agent reaches the target, consider re-solving the now discovered gridworld for the shortest path (eliminating any backtracking that may have occurred). Will this be an optimal path in the complete gridworld? Argue for, or give a counter example.

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4: Solvability A gridworld is solvable if it has a clear path from start to goal nodes. How does solvability depend on p? Given dim = 101, how does solvability depend on p? For a range of p values, estimate the probability that a maze will be solvable by generating multiple environments and checking them for solvability. Plot density vs solvability, and try to identify as accurately as you can the threshold p0 where for p < p0, most mazes are solvable, but p > p0, most mazes are not solvable. Is A* the best search algorithm to use here, to test for solvability? Note for this problem you may assume that the entire gridworld is known, and hence only needs to be searched once each.

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5: Performance Taking dim = 101, for a range of density p values from 0 to min(p0, 0.33), and the heuristic chosen as best in Q5, repeatedly generate gridworlds and solve them using Repeated Forward A*. Use as the field of view each immediately adjacent cell in the compass directions. Generate plots of the following data:

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Density vs Average Trajectory Length

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Ã¯Â¿Â½Density vs Average (Length of Trajectory / Length of Shortest Path in Final Discovered Gridworld)

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Density vs Average (Length of Shortest Path in Final Discovered Gridworld / Length of Shortest Path in Full Gridworld)

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Density vs Average Number of Cells Processed by Repeated A*

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Discuss your results. Are they as you expected? Explain.

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6: Performance Part 2 Generate and analyze the same data as in Q6, except using only the cell in the direction of attempted motion as the field of view. In other words, the agent may attempt to move in a given direction, and only discovers obstacles by bumping into them. How does the reduced field of view impact the performance of the algorithm? The final questions asks you to be somewhat speculative and explore (you may pick one, doing both will be viewed as extra credit). Both require code and data to answer completely:

7: Improvements Repeated A* may suffer in that the best place to re-start A* from may not be where you currently are - for instance if you are at the dead end of a long hallway, you can save some effort by backtracking to the end of the hallway (recycling information you already have) before restarting the A* search. By changing where you restart the search process, can you cut down the overall runtime? What effect does this have on the overall trajectory (given that you have to travel between the current position and the new initial search position)?

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8: Heuristics A* can frequently be sped up by the use of inadmissible heuristics - for instance weighted heuristics or combinations of heuristics. These can cut down on runtime potentially at the cost of path length. Can this be applied here? What is the effect of weighted heuristics on runtime and overall trajectory? Try to reduce the runtime as much as possible without too much cost to trajectory length.