chapter 10: metric path planning a. representations b. algorithms

10Chapter 10:

Metric Path Planning

a. Representationsb. Algorithms

Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning 2

10 Objectives

• Define Cspace, path relaxation, digitization bias, subgoal obsession, termination condition

• Explain the difference between graph and wavefront planners

• Represent an indoor environment with a GVG, a regular grid, or a quadtree, and create a graph suitable for path planning

• Apply A* search

• Apply wavefront propagation

• Explain the differences between continuous and event-driven replanning


10 Navigation• Where am I going? Mission

planning

• What’s the best way there? Path planning

• Where have I been? Map making

• Where am I? Localization

MissionPlanner

Carto-grapher

BehaviorsBehaviorsBehaviorsBehaviors

deli

bera

tive

reac

tiveHow am I going to get

there?


10 Spatial Memory

• What’s the Best Way There? depends on the representation of the world

• A robot’s world representation and how it is maintained over time is its spatial memory– Attention

– Reasoning

– Path planning

– Information collection

• Two forms– Route (or qualitative)

– Layout (or metric)

• Layout leads to Route, but not the other way


10 Representing Area/Volume in Path Planning

• In Quantitative or metric maps:– Rep: Many different ways to represent an area or volume

of space• Looks like a “bird’s eye” view, position & viewpoint

independent

– Algorithms• Graph or network algorithms

• Wavefront or graphics-derived algorithms


10 Metric Maps• Motivation for having a metric map is often path

planning (others include reasoning about space…)

• Determine a path from one point to goal– Generally interested in “best” or “optimal”

– What are measures of best/optimal?

– Relevant: occupied or empty

• Path planning assumes an a priori map of relevant aspects– Only as good as last time map was updated


10 Metric Maps use Cspace• World Space: physical space robots and obstacles exist in

– In order to use, generally need to know (x,y,z) plus Euler angles: 6DOF

• Ex. Travel by car, what to do next depends on where you are and what direction you’re currently heading

• Configuration Space (Cspace)– Transform space into a representation suitable for robots,

simplifying assumptions

6DOF 3DOF


10 Major Cspace Representations

• Idea: reduce physical space to a cspace representation which is more amenable for storage in computers and for rapid execution of algorithms

• Major types

– Meadow Maps

– Generalized Voronoi Graphs (GVG)

– Regular grids, quadtrees


10 Meadow Maps

• Example of the basic procedure of transforming world space to cspace

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


10 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”)

– Oops, not necessarily unique set of polygons


10 Meadow Maps cont.• Step 3: represent convex polygons in way suitable for path

planning-convert to a relational graph– Is this less storage, data points than a pixel-by-pixel representation?


10 Problems with Meadow Maps

• Not unique generation of polygons

• Could you actually create this type of map with sensor data?

• How does it tie into actually navigating the path?– How does robot recognize “right” corners, edges and go to

“middle”?

– What about sensor noise?


10 Path Relaxation

• Get the kinks out of the path– Can be used with any cspace representation


10 Class Exercise

• Create a meadow map and relational graph using mid-point of line segments


10 Generalized Voronoi Graphs • Imagine a fire starting at the boundaries, creating a line where they

intersect, intersections of lines are nodes

• Result is a relational graph


10 Class Exercise

• Create a GVG of this space


10 Regular Grids

• Bigger than pixels, but same idea– Often on order of 4inches square

– Make a relational graph by each element as a node, connecting neighbors (4-connected, 8-connected)

– Moore’s law effect: fast processors, cheap hard drives, who cares about overhead anymore?


10 Class Exercise

• Use 4-connected neighbors to create a relational graph


10 Problems with GVG and Regular Grids

• GVG– Sensitive to sensor noise

– Path execution: requires robot to be able to sense boundaries

• Grids– World doesn’t always line up on grids

– Digitalization bias: left over space marked as occupied


10 Summary of Representations• Metric path planning requires

– Representation of world space, usually try to simplify to cspace– Algorithms which can operate over representation to produce

best/optimal path

• Representation– Usually try to end up with relational graph– Regular grids are currently most popular in practice, GVGs are

interesting– Tricks of the trade

• Grow obstacles to size of robot to be able to treat holonomic robots as point

• Relaxation (string tightening)

• Metric methods often ignore issue of– how to execute a planned path– Impact of sensor noise or uncertainty, localization


10 Algorithms

• For Path planning– A* for relational graphs

– Wavefront for operating directly on regular grids

• For Interleaving Path Planning and Execution


10 Motivation for A*

• Single Source Shortest Path algorithms are exhaustive, visiting all edges– Can’t we throw away paths when we see that they aren’t going

to the goal, rather than follow all branches?

• This means having a mechanism to “prune” branches as we go, rather than after full exploration

• Algorithms which prune earlier (but correctly) are preferred over algorithms which do it later.

• Issue -> the mechanism for pruning


10 A*• Similar to breadth-first: at each point in the time the

planner can only “see” it’s node and 1 set of nodes “in front”

• Idea is to rate the choices, choose the best one first, throw away any choices whenever you can

• f*(n) is the “goodness” of the path from Start to n• g*(n) is the “cost” of going from the Start to node n• H*(n) is the cost of going from n to the Goal

– H is for “heuristic function” because must have a way of guessing the cost of n to Goal since can’t see the path between n and the Goal

f*(n)=g*(n)+h*(n)


10 A* Heuristic Function

• G*(n) is easy: just sum up the path costs to n

• H*(n) is tricky– But path planning requires an a priori map

– Metric path planning requires a METRIC a priori map

– Therefore, know the distance between Initial and Goal nodes, just not the optimal way to get there

– H*(n)= distance between n and Goal

f*(n)=g*(n)+h*(n)


10 Example: A to E

• But since you’re starting at A and can only look 1 node ahead, this is what you see:

AB

D

F E

1

1

1

1

1.4

1.4

AB

D

E

1

1.4


10

• Two choices for n: B, D

• Do both– f*(B)=1+2.24=3.24

– f*(D)=1.4+1.4=2.8

• Can’t prune, so much keep going (recurse)– Pick the most plausible path first => A-D-?-E

AB

D

E

1

1.4

1.4

2.24


10• A-D-?-E

– “stand on D”– Can see 2 new nodes: F, E– f*(F)=(1.4+1)+1=3.4– f*(E)=(1.4+1.4)+0=2.8

• Three paths– A-B-?-E >= 3.24– A-D-E = 2.8– A-D-F-?-D >=3.4

• A-D-E is the winner! – Don’t have to look farther because expanded the shortest first, others couldn’t possibly do

better without having negative distances, violations of laws of geometry…

AB

D

E

1

1.4

1.4

F1

1


10 Class Exercise

• Find the best path from A to E

D

F E

1

1

1

1

1.4

1.4

AB

C


10 Wavefront Planners


10 Trulla


10 Interleaving Path Planning and Reactive Execution

• Graph-based planners generate a path and subpaths or subsegments

• Recall NHC, AuRA– Pilot looks at current subpath, instantiates behaviors to get

from current location to subgoal

• What happens if blocked? What happens if avoid an obstacle and actually is now closer to the next subgoal?– Causes Subgoal obsession

• When does the robot think it has reached (x,y)? What about encoder error? – Need a Termination condition, usually +/- width of robot


10 Trulla Example


10 Two Example Approaches

• If compute all possible paths in advance, not a problem– Shortest path between pairs will be part of

shortest path to more distant pairs

• D* – Tony Stentz

– Run A* over all possible pairs of nodes

• Trulla– Ken Hughes

– By-product of wave propagation style is path to everywhere


10 Summary• Metric path planners

– graph-based (A* is best known)

– Wavefront

• Graph-based generate paths and subgoals.– Good for NHC styles of control

– In practice leads to:• Subgoal obsession

• Termination conditions

• Planning all possible paths helps with subgoal obsession– What happens when the map is wrong, things change, missed

opportunities? How can you tell when the map is wrong or that’s it worth the computation?

chapter 10: metric path planning a. representations b. algorithms

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