Chapter 10: Metric Path Planning a. Representations b. Algorithms

1 / 29

# Chapter 10: Metric Path Planning a. Representations b. Algorithms - PowerPoint PPT Presentation

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 &amp; viewpoint independent Algorithms

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Chapter 10: Metric Path Planning a. Representations b. Algorithms' - melvin-taylor

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Chapter 10:Metric Path Planninga. Representationsb. 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

Graph or network algorithms

Wavefront or graphics-derived algorithms

Chapter 10: Metric Path Planning

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
• 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
• Idea: reduce physical space to a cspace representation which is more amenable for storage in computers and for rapid execution of algorithms
• Major types
• Generalized Voronoi Graphs (GVG)

Chapter 10: Metric Path Planning

• 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

• 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

• 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

• 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”?

Chapter 10: Metric Path Planning

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

Chapter 10: Metric Path Planning

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
• 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
• 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
• 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
• 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
• 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*
• 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
• 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

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

Chapter 10: Metric Path Planning

Trulla

Chapter 10: Metric Path Planning

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
• 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 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

Chapter 10: Metric Path Planning

Summary
• Metric path planners
• graph-based (A* is best known)
• Wavefront
• Graph-based generate paths and subgoals.
• Good for NHC styles of control
• 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:
• 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