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Chapter 10: Metric Path Planning a. Representations b. Algorithms. Representing Area/Volume in Path Planning. Quantitative or metric Rep: Many different ways to represent an area or volume of space Looks like a “bird’s eye” view, position & viewpoint independent Algorithms

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Chapter 10 metric path planning a representations b algorithms

Chapter 10:Metric Path Planninga. Representationsb. Algorithms


Representing area volume in path planning
Representing Area/Volume in Path Planning

Quantitative or metric

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

Chapter 10: Metric Path Planning


Metric maps
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

Chapter 10: Metric Path Planning


Metric maps use cspace
Metric Maps use Cspace

  • World Space: physical space robots and obstacles existin

    • 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

Chapter 10: Metric Path Planning


Major cspace representations
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

Chapter 10: Metric Path Planning


Meadow maps
Meadow Maps

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

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

Chapter 10: Metric Path Planning


Meadow maps cont
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

Chapter 10: Metric Path Planning


Meadow maps cont1
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?

Chapter 10: Metric Path Planning


Problems with meadow maps
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?

Chapter 10: Metric Path Planning


Path relaxation
Path Relaxation

  • Get the kinks out of the path

    • Can be used with any cspace representation

Chapter 10: Metric Path Planning


Generalized voronoi graphs
Generalized Voronoi Graphs

  • Create lines equidistant from objects and walls,

  • Intersections of lines are nodes

  • Result is a relational graph

Chapter 10: Metric Path Planning


Regular grids
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?

Chapter 10: Metric Path Planning


Problems with gvg and regular grids
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

Chapter 10: Metric Path Planning


Summary
Summary

  • 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

Chapter 10: Metric Path Planning


Algorithms
Algorithms

  • Path planning

    • A* for relational graphs

    • Wavefront for operating directly on regular grids

  • Interleaving Path Planning and Execution

Chapter 10: Metric Path Planning


Motivation for a
Motivation for A*

  • Single Source Shortest Path algorithms are exhaustive, visting 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

Chapter 10: Metric Path Planning


A*

  • Similar to breadth-first: at each point of 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 “cost” of the path from Start to Goal through node n

  • g*(n) is the “cost” of going from Start to node n

  • h*(n) is the cost of going from n to the Goal

    • h* is a “heuristic function” because it must have a way of guessing the cost of n to Goal since it can’t see the path between n and the Goal

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

Chapter 10: Metric Path Planning


A heuristic function
A* Heuristic Function

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

  • h*(n) is tricky

    • 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

    • h*(n) <= h(n)

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

Chapter 10: Metric Path Planning


Example a to e
Example: A to E

1

F

E

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

1

1.4

D

1.4

1

1

B

A

E

D

1.4

1

B

A

Chapter 10: Metric Path Planning


E

1.4

2.24

D

  • 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

1.4

1

B

A

Chapter 10: Metric Path Planning


1

E

F

1

1.4

D

  • 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…

1.4

1

B

A

Chapter 10: Metric Path Planning


Wavefront planners
Wavefront Planners

Chapter 10: Metric Path Planning


Trulla
Trulla

Chapter 10: Metric Path Planning


Interleaving path planning and reactive execution
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

  • When the robot tries to reach a subgoal, it may exhibit subgoal obsession due to an encoder error - it is necessary to allow a tolerance corresponding usually to +/- width of robot

  • What happens if a goal is blocked? - need a Termination condition, e.g. deadline

  • What happens if a robot avoiding an obstacle is now closer to the next subgoal? - it would be good to use an opportunistic replanning

Chapter 10: Metric Path Planning


Two example approaches
Two Example Approaches

  • If computing all possible paths in advance, there not a problem

    • Shortest path between pairs will be part of shortest path to more distant pairs

  • D*

    • Run A* over all possible pairs of nodes

    • continuously update the map

    • disadvantages: 1. too computationally expensive to be practical for a robot 2. continuous replanning is highly dependent on sensing quality,

Chapter 10: Metric Path Planning


Two example approaches1
Two Example Approaches

  • Event driven scheme - event noticeable by a reactive system would trigger replanning

  • the Trulla planner uses for this dot product of the intended path vector and the actual path vector

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

  • for opportunistic replanning in case of favorable change D* is better, because it will automatically notice the change, while Trulla will not notice it

Chapter 10: Metric Path Planning


Trulla example
Trulla Example

Chapter 10: Metric Path Planning


Summary1
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


You should be able to
You should be able to:

  • 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

Chapter 10: Metric Path Planning


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