Artificial Intelligence: Methods and applications Lecture 6: Path - - PowerPoint PPT Presentation

Artificial Intelligence: Methods and applications

Lecture 6: Path planning Ola Ringdahl Umeå University November 21, 2014

Navigation

Four questions:

• Where am I going?

– Mission planning (human or planner)

• What is the best way to get there?

– Path planning (Topological or Metric)

• Where have I been?

– Map making

• Where am I now?

– Localization (relative or absolute)

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Spatial Memory

• The answer to “What’s the Best Way There?”

depends on the representation of the world

• A robot’s world representation is its spatial
• memory. It support the following functions:

– Attention: What features to look for next? – Reasoning: Can this surface support my weight? – Path planning: What’s the best way? – Information collection: What has changed since last time I was here?

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Two forms of Spatial Memory

Based on landmarks (qualitative/Topological) Based on regular maps (quantitative/Metric)

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Topological Spatial Memory

• Express space in terms of

connections between landmarks

• From the robot’s perspective

– Identification of landmarks – Orientation clues, e.g. “to the left”

• Usually cannot be used to generate metric

representations

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Metric Spatial Memory

• Express space in terms of

physical distances of travel: A metric map

• Bird’s eye view of the world
• Not dependent upon the perspective of the

robot

– Independent of orientation and position of robot

• Can also be used to generate topological

representations

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USE A TOPOLOGICAL SPATIAL MEMORY

• 9. TOPOLOGICAL PATH

PLANNING

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Landmarks

One or more perceptually distinctive features of interest on an object or area of interest

• Natural landmark: wasn’t put in

the environment to aid with the robot’s navigation (e.g. tower, corner, tree, doorway)

• Artificial landmark: added to the

environment to support navigation (e.g. highway sign, RFID tags) Try to avoid artificial landmarks!

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Desirable Characteristics

• f Landmarks
• Recognizable
• From sufficiently long range
• From different viewpoints
• Supply necessary information
• Identity (unique globally or at least locally)
• Relative orientation and distance to the

landmark

• …

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Two types of Topological navigation methods

• Relational
• spatial memory (known as a topological map
• r relational graph) is based on landmarks
• use graph theory to plan paths
• Associative
• spatial memory is a series of remembered

viewpoints, where each viewpoint is labeled with a location

• good for retracing steps

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Constructing a Topological map

• Draw edges and nodes to cover the area
• Nodes:

– possible goals or gateways (where the direction may change)

• Edges:

– Navigable paths between nodes

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Room 1 Room 2 Room 3 Room 4

Artificial Intelligence: Methods and applications Ola Ringdahl, Umeå University

Problems with relational graphs

• The graph is not coupled with information on

how to get from one node to another

• Dead reckoning accumulates uncertainty

– Possible solution: add localization to landmarks

• Hard to find good distinctive places

(features/landmarks)

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Topological Path Planning Algorithms

• Answers “What’s the best way there?”
• Find a sequence of nodes that leads you to

the goal!

• Relational graph, so any shortest path

algorithm will work, e.g. Dijkstra’s algorithm

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Two types of Topological navigation methods

• Relational
• spatial memory is based on landmarks (also

known as a topological map or relational graph)

• use graph theory to plan paths
• Associative
• spatial memory is a series of remembered

viewpoints, where each viewpoint is labeled with a location

• good for retracing steps (Path Tracking)
• converts sensor observations to direction

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Associative Method example 1

• Visual Homing
• bees navigate to their hive by a series of

image signatures which are locally distinctive (within a neighborhood)

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Artificial Intelligence: Methods and applications Ola Ringdahl, Umeå University

Image Signatures for Visual Homing

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The world Tessellated (like faceted-eyes) Resulting signature for home

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Image Signatures for Visual Homing

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Move to match the template

Associative Method example 2

• QualNav
• Developed as a military project
• The UGV had to check an area and return

home without being seen: i.e: had to move far away from potential landmarks

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QualNav

• Works out-doors with landmarks far away
• Landmark pair boundary: Imaginary line

drawn between 2 landmarks. Partitions the world into Orientation regions (OR).

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mountain tree building radio tower

OR1 OR2

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QualNav

• Within an OR: to localize the robot, use recorded

angles to the landmarks (viewframe)

• When the robot moves from one OR to another,

the border Landmark pair boundaries will move in front /on the side/behind. No distances needed.

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mountain tree building radio tower

OR1 OR2

Metric Map Topological map

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Summary Topological Path Planning

• Navigating by detecting and responding to

landmarks.

• Landmarks may be natural or artificial
• Two types of topological path planning

– Relational – Associative

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Summary Topological Path Planning

• Relational methods

– spatial memory is based on landmarks – use graph theory to plan paths in the topological map

• Associative methods

– remember places as image signatures or as extracted viewframes – direct stimulus-response coupling by matching perception to signature to response – Assume perceptual stability and perceptual distinguishability – Sensitive to changes in the world

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USE A METRIC SPATIAL MEMORY

10 METRIC PATH PLANNING

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Metric Path Planning

• Determine a path from one point to goal

– Generally interested in “best” or “optimal” – What are measures of best/optimal?

• Path planning assumes an a priori map of

relevant aspects

– Relevant: occupied or empty – Looks like a “bird’s eye” view, position & viewpoint independent

• We will look at Representations and

Algorithms

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Artificial Intelligence: Methods and applications Ola Ringdahl, Umeå University

REPRESENTATIONS FOR METRIC PATH PLANNING

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6DOF 2 or 3 DOF

World Space & Cspace

• World Space: physical space robots and
• bstacles exist in

– (x,y,z) plus three angles: 6DOF.

• Configuration Space (Cspace)

– A transformation into a representation suitable for planning, simplifying assumptions, e.g. fewer DOFs: – Or considering only a limited number of poses (grids and graphs)

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Cspace Representations

• Idea: reduce world space to a Cspace

representation which is more suitable for storage in computers and for rapid execution of path planning algorithms

• Major types

– Meadow Maps – Generalized Voronoi Graphs (GVG) – Regular grids, quadtrees

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Meadow Maps

• Example of the basic procedure of

transforming world space to cspace

• Step 1 (optional): grow obstacles as big as

robot

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Meadow Maps cont.

• Step 2: Construct convex polygons as line

segments between pairs of corners, edges

– Why convex polygons? Interior has no obstacles so can safely transit (“freeway”, “free space”)

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Meadow Maps cont.

• Step 3: Convert to a relational graph by

connecting midpoints of lines of the polygons

• A search algorithm can now find the ”optimal”

path

Problems with Meadow Maps

• How does it tie into actually navigating the

path?

• Does it make sense to move to one midpoint

and then turn towards another midpoint

• Often jagged paths
• Could you actually create this type of map

with sensor data?

• What about sensor noise?
• How does robot recognize “right” corners and

edges

• Computationally complex

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Artificial Intelligence: Methods and applications Ola Ringdahl, Umeå University 33

Path Relaxation

• Step 4: Get the kinks out of the path

– Can be used with any cspace representation

Generalized Voronoi Graphs

• A line (Voronoi edge) is formed by points

that are equidistant from two or more

• bstacles
• Intersections of lines is a node (Voronoi

vertex)

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Generalized Voronoi Graphs

• Result is a relational graph
• A search algorithm can now find the

”optimal” path

• No need to grow obstacles as the robot

stays “in the middle”

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Generalized Voronoi Graphs

• Easy to follow the edges, e.g. “follow

corridor” (can be used to actually create Voronoi diagrams of unknown environments)

• Sensitive to sensor noise

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Regular Grids

• Like an occupancy grid
• ~10cm squares to avoid digitization bias
• First: grow obstacles
• Use the grid as Cspace or make a relational

graph with each element as a node (4- connected or 8-connected)

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Problems with Regular Grids

• Large search graphs. But:

– Moore’s law, fast processors, cheap hard drives, .. – Who cares about overhead anymore?

• World doesn’t always line up on grids

– Digitization bias: almost empty cells are marked as occupied – Can lead to jagged paths

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Quadtrees

• If an object falls into only a part of a cell:

divide the cell into 4 new cells

– Do this recursively till all obstacles fills whole cells

• Avoids digitation bias

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Summary of metric representations

• Metric path planning require

– A simplified Cspace representation of world space – Algorithms which can produce best/optimal paths in Cspace

• Representation

– Usually try to end up with relational graph (Meadow maps, GVGs, Regular grids) – Tricks of the trade

• Grow obstacles to size of robot to be able to treat a

holonomic robot as a point

• Relaxation (string tightening)
• Metric methods often ignore issue of

– how to identify nodes in the real world – impact of sensor noise or uncertainty, localization

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ALGORITHMS FOR METRIC PATH PLANNING

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Algorithms

Most Cspace representations can be converted to relational graphs with weighted edges. We need algorithms for:

• Path planning

– A* for relational graphs – Wavefront for operating directly on regular grids

• Interleaving Path planning and Path tracking

(execution)

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Wavefront Planners

• Well-suited for grid representations
• General idea: consider Cspace to be

conductive material with heat radiating out from initial node to goal node

• If there is a path, heat will eventually reach

goal node

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Artificial Intelligence: Methods and applications Ola Ringdahl, Umeå University

Wavefront Planners

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Start Goal

Wavefront Planners

• For each cell: Mark the direction from where

the heat came

– E.g. by assigning increasing numbers to cells as the wave propagates

• Results in a map that looks like a potential

field

• Optimal path from all grid elements to the

goal can be computed

• Can handle different terrains:
• Obstacle: zero conductivity
• Free space: infinite conductivity
• Undesirable terrains: low conductivity

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Optimal path generated by the Wavefront Planner

Start Goal

Interleaving Path Planning and Path Tracking

• A Graph-based planner generate a path and

path segments

• The path tracker looks at current path

segment and instantiates behaviors to get from current location to a (sub)goal

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Interleaving Path Planning and Path Tracking

• Problem: Subgoal obsession (when has the

robot reached the subgoal (x,y) )?

• Solutions:

– Termination condition: usually +/- width of robot – Compute all possible paths in advance

• D*: Run A* from all possible nodes to the

goal

• Wavefront path planning - gives path from

everywhere (the map is a “virtual sensor”)!

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Interleaving Path Planning and Path Tracking

• Problem: Incorrect map – unmodeled or

new obstacles drive the robot closer to another subgoal

• Solutions:

– Opportunistic replanning (also helps against subgoal obsession) – Compute all possible paths in advance (E.g. D* see previous slide)

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Summary metric path planning

• Use obstacle growing to enable treating

robot as a point

• Graph-based planners (A* is best known)

– Generate paths and subgoals – In practice may lead to subgoal obsession and a need for opportunistic replanning or preplanning

• Wavefront based planners

– Work well with regular grids – Built in opportunistic replanning – Can easily account for varying traversability (sand, rocky terrain, carpets, crowds, etc)

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