graphs david johnson. describing the environment how do you tell a person/robot how to get...
TRANSCRIPT
Graphs
David Johnson
Describing the Environment
• How do you tell a person/robot how to get somewhere?– Tell me how to get to the student services building…
World Representation• Landmarks• Paths/Roads• Map
• This is abstracted as a graph– Not in the sense of a plot
Graphs• A graph is a set of
landmarks and the paths between them
• A graph G has– Set of vertices (or nodes) V– Set of edges E
• A graph is a classic data structure in CS
V1
V2
V3
E1
E2
Graphs
• A directed graph means – E(A,B) does not imply
E(B,A)– What kind of robot would
need a graph that is not bidirectional?
• Edges may have weights– Where would this be
appropriate?
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V3
E1
E2
1.4
2.6
Paths
• A path is a sequence of vertices such that for Vi and Vi+1 Ei,i+1 exists
• A graph is connected if there is a path between all V– For what kind of robot and
environment would this not be true?
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V2
V3
E1
E2
1.4
2.6
Imagine a simple robot• What is the simplest
kind of robot you can think of?
Point Robots
• A point robot can move along the infinitesimal width edges of a graph.
• Path planning for this simple case is just searching a graph for a path.
Finding Paths
• Given a start and end, how do we find a path between them? V1
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Depth First Search (DFS)
• Always take the first option
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Depth First Search (DFS)
• Always take the first option
V1
V2
V1V3
V4
V7
V6
V8
V5
Depth First Search (DFS)
• Always take the first option
V1
V2
V1V3
V4
V7
V6
V8
V5
Depth First Search (DFS)
• Always take the first option
V1
V2
V1V3
V4
V7
V6
V8
V5
Depth First Search (DFS)
• Always take the first option
V1
V2
V1V3
V4
V7
V6
V8
V5
Depth First Search (DFS)
• Always take the first option
V1
V2
V1V3
V4
V7
V6
V8
V5
Depth First Search (DFS)
• Always take the first option
V1
V2
V1V3
V4
V7
V6
V8
V5
Breadth First Search (BFS)
• Exhaust all possibilities at same level first V1
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V1V3
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V5
Breadth First Search (BFS)
• Exhaust all possibilities at same level first V1
V2
V1V3
V4
V7
V6
V8
V5
Breadth First Search (BFS)
• Exhaust all possibilities at same level first V1
V2
V1V3
V4
V7
V6
V8
V5
Breadth First Search (BFS)
• Exhaust all possibilities at same level first V1
V2
V1V3
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V5
Representing a graph in Matlab
• Demo
Wavefront planner
• Use BFS on a grid• Label cells with values
– Start with zero– Expand from start – Add +1 to neighbors of current wavefront– Use gradient descent to search from goal to
start
Representations: A Grid
• Distance is reduced to discrete steps– For simplicity, we’ll assume distance is uniform
• Direction is now limited from one adjacent cell to another
Representations: Connectivity
• 8-Point Connectivity– (chessboard metric)
• 4-Point Connectivity– (Manhattan metric)
The Wavefront Planner: Setup
The Wavefront in Action (Part 1)
• Starting with the goal, set all adjacent cells with “0” to the current cell + 1– 4-Point Connectivity or 8-Point Connectivity?– Your Choice. We’ll use 8-Point Connectivity in our example
The Wavefront in Action (Part 2)
• Now repeat with the modified cells– This will be repeated until goal is reached
• 0’s will only remain when regions are unreachable
The Wavefront in Action (Part 3)
• Repeat again...
The Wavefront in Action (Part 4)
• And again...
The Wavefront in Action (Part 5)
• And again until...
The Wavefront in Action (Done)
• You’re done
The Wavefront
• To find the shortest path simply always move toward a cell with a lower number– The numbers generated by the Wavefront planner are roughly
proportional to their distance from the goal
Twopossibleshortestpathsshown