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# 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 & viewpoint independent Algorithms

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

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

• 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

• 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

• 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

• 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”?

• What about sensor noise?

Chapter 10: Metric Path Planning

• Get the kinks out of the path

• Can be used with any cspace representation

Chapter 10: Metric Path Planning

• Create lines equidistant from objects and walls,

• Intersections of lines are nodes

• Result is a relational graph

Chapter 10: Metric Path Planning

• 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

• 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

• 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

• Path planning

• A* for relational graphs

• Wavefront for operating directly on regular grids

• Interleaving Path Planning and Execution

Chapter 10: Metric Path Planning

• 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

• 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

• 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

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

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

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

Chapter 10: Metric Path Planning

Chapter 10: Metric Path Planning

• 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

• 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

• 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

Chapter 10: Metric Path Planning

• 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

• 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