Algorithms and Path Planning

Transcription

1 Algorithms and Path Planning Topics Simple Search Depth First Search Breadth First Search Dijkstra s Search Greedy Search A* Search Classes of interest ECE400: Computer Systems Programming CS4700: Foundations of Artificial Intelligence CS470: Practicum of Artificial Intelligence CS3758/MAE480: Autonomous Mobile Robots ECE 3400: Intelligent Physical Systems

2 End Game 5 weeks left till competition day 7Ups Can you explore the entire maze? 5s avg. for 6 squares 3.4min for 8 squares Unlikely, but possible.

3 Algorithms and Search Depth First Search (DFS) Advantage? Disadvantage? (0,0) (0,) (,0) (0,) (,) Search order: N, E, S, W Find a treasure (,4) (0,3) (,) (0,4) (,3) Memory grows linearly with the depth of the graph and so on y S x

4 Algorithms and Search Depth First Search (DFS) Breadth First Search (BFS) (0,) (0,) (,) (0,0) (,) Advantage? Disadvantage? (,0) (,0) Find a treasure 3 Search order: N, E, S, W (0,3) (,) (,) (,)(,) (,) (,) (3,0) 4 and so on Memory grows exponentially with the depth of the graph y S 5 x

5 BFS: Memory and Computation What do we need? Locations Example issue from last year n = state(init) frontier.append(n) while(frontier not empty) n = pull state from frontier if n = goal, return solution for all actions in n n = a(n) frontier.append(n ) nodes explored BFS: Growth of frontier Arduino no go! R depth (distance from robot) 5

6 BFS: Memory and Computation What do we need? Locations Parents Frontier size: 4 36 etc mem = 4۰3 n- (n = depth) 0000 BFS: Growth of frontier 500 BFS: Growth of frontier R Don t revisit parents/grandparents nodes explored 4000 nodes explored depth depth 6

7 BFS: Memory and Computation What do we need? Locations Parents Visited What is the maximum size of the frontier now? nodes explored BFS: Growth of frontier What is the issue with this approach? Store branches with lowest motion cost! R depth 7

8 BFS: Memory and Computation What do we need? Locations* Parents* Visited Cost* Action* n = state(init) frontier.append(n) visited.append(n) while(frontier not empty) * n = pull state from frontier if n = goal, return solution for all actions in n n = a(n) if n not visited or cost is lower frontier.append(n ) visited.append(n ) R 8

9 BFS: Memory and Computation visited? treasure robot? N,E,S,W walls? visited? treasure N,E,S,W walls? R Teams are at 75-35% capacity (5B-33B) Frontier (x,y-locations + parent + cost): 80B Visited: 8B, or 0B! Maze: 6B Maze: 8B 9

10 Maze Exploration Depth First Search Search order: Straight Left, then straight Right, then straight U-turn, then straight If stuck, find shortest path to the next frontier in the tree What treasure does the robot find first? R R 0

11 Algorithms and Search Depth First Search (DFS) Breadth First Search (BFS) Add motion cost Dijkstra s to save computation/memory (0,) (0,0) (,0) Find a treasure 3 Search order: N, E, S, W (0,) (,) (,) (,0) 4 (0,3) (,) (,) (,)(,) (,) (,) (3,0) and so on y S 5 x

12 Algorithms and Search Depth First Search (DFS) Breadth First Search (BFS) Add motion cost Dijkstra s to save computation/memory (0,) (0,0) (,0) Find a treasure 3 Search order: N, E, S, W (0,) (,) (,) (,0) 4 (0,3) (,) and so on y S 5 x

13 Maze Exploration Depth First Search Search order: Straight Left, then straight Right, then straight U-turn, then straight If stuck, find shortest path to the next frontier in the tree What treasure does the robot find first? What treasure does the robot find second? Could we be more efficient? R R R 3

14 Maze Exploration Dijkstra to find the next frontier Search order: Straight Left, then straight Right, then straight U-turn, then straight What treasure does the robot find first? Next treasure? Extra computation (Dijkstra for every square), but maybe better Better path planning? Add cost for revisiting nodes R

15 Maze Exploration Dijkstra s Search Could you be more efficient by looking ahead? (0,) (0,0) Only reasons about the cost to get there (,0) (0,) + (,) (,0) Find a treasure 5 Search order: N, E, S, W (0,3) + + (,) (,) (3,0) and so on y S 3 4 x

16 Informed Search Greedy Search (0,3) (0,4) (0,) (0,) (,) (,) (,3) (,4) 0(,3) (0,0) 4 4 (,0) Define a heuristic to target the goal Manhatten distance abs(x S -x G )+abs(y S -y G ) y Find a treasure 3 4 S x Search order: N, E, S, W

17 Informed Search Greedy Search Cause for concern? Find a treasure Search order: N, E, S, W n = state(init) frontier.append(n) while(frontier not empty) n = pull state from frontier visited.append(n) if n = goal, return solution for all actions in n n = a(n) if n not visited priority = heuristic(goal,n ) frontier.append(priority) y 3 4 S x

18 Informed Search Greedy Search Cause for concern? Faster, but does not guarantee optimal Find a treasure Search order: N, E, S, W y S x

19 Informed Search Breadth First Search Guarantee: Finds a path Searches everything A* Dijkstra s Algorithm Considers parent cost Guarantee: Finds the shortest path but it wastes time exploring in directions that aren t promising Greedy Search Considers goal Guarantee: Finds a path only explores promising directions

20 Informed Search A* ( A-star ) n = state(init) frontier.append(n) while(frontier not empty) n = pull state from frontier if n = goal, return solution for all actions in n n = a(n) if ((n not visited or (visited and n.cost < n_old.cost)) priority = heuristic(goal,n )+cost frontier.append(priority) visited.append(n ) Find a treasure S Search order: N, E, S, W

21 Informed Search A* ( A-star ) Cost and goal heuristic (0,0) (0,) (,0) (0,) (,) (,0) (0,3) (,) (,) 4 (3,0) 3 (0,4) 3 (,3) (,4) (,) (3,) (,3) goal (3,) y Search order: N, E, S, W Find a treasure S 5 x

22 Informed Search What if the heuristic is too optimistic? Estimated cost < true cost What if the heuristic is too pessimistic? Estimated cost > true cost No longer guaranteed to be optimal What if the heuristic is just right? Pre-compute the cost between all nodes Feasible for you? admissible heuristic inadmissible heuristic

23 Summary Dijkstra minimum path Greedy A* minimum path and efficient S S S 5 3

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25 Game Theory Pick a whole number between and 00. The winner is the person who picks the value which is closest to two thirds of the class average. E.g. [0, 0, 60]. Class average 30. Winner: 0. The poll will close at the end of the class (.0pm 0/9 th ) 5

26 Class website: Piazza: 6