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

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Chapter 10Metric Path Planninga.
Representationsb. Algorithms
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Representing Area/Volume in Path Planning

• Quantitative or metric
• Rep Many different ways to represent an area or
  volume of space
• Looks like a birds eye view, position
  viewpoint independent
• Algorithms
• Graph or network algorithms
• Wavefront or graphics-derived algorithms

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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

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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 youre currently
  heading
• Configuration Space (Cspace)
• Transform space into a representation suitable
  for robots, simplifying assumptions

6DOF
3DOF
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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

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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)
• Oops, not necessarily unique set of polygons

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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?

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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?

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Path Relaxation

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

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Generalized Voronoi Graphs

• Create lines equidistant from objects and walls,
• Intersections of lines are nodes
• Result is a relational graph

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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)
• Moores law effect fast processors, cheap hard
  drives, who cares about overhead anymore?

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Problems with GVG and Regular Grids

• GVG
• Sensitive to sensor noise
• Path execution requires robot to be able to
  sense boundaries
• Grids
• World doesnt always line up on grids
• Digitalization bias left over space marked as
  occupied

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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

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Algorithms

• Path planning
• A for relational graphs
• Wavefront for operating directly on regular grids
• Interleaving Path Planning and Execution

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Motivation for A

• Single Source Shortest Path algorithms are
  exhaustive, visting all edges
• Cant we throw away paths when we see that they
  arent 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

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A

• Similar to breadth-first at each point of time
  the planner can only see its 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
  cant see the path between n and the Goal

f(n)g(n)h(n)
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A Heuristic Function
f(n)g(n)h(n)

• 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) lt h(n)

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Example A to E
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F
E
1
1.4
D
1.4
1
1
A
B

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

E
D
1.4
1
A
B
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E
1.4
2.24
D
1.4
1
A
B

• Two choices for n B, D
• Do both
• f(B)12.243.24
• f(D)1.41.42.8
• Cant prune, so much keep going (recurse)
• Pick the most plausible path first A-D-?-E

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1
E
F
1
1.4
D
1.4
1
A
B

• A-D-?-E
• stand on D
• Can see 2 new nodes F, E
• f(F)(1.41)13.4
• f(E)(1.41.4)02.8
• Three paths
• A-B-?-E gt 3.24
• A-D-E 2.8
• A-D-F-?-D gt3.4
• A-D-E is the winner!
• Dont have to look farther because expanded the
  shortest first, others couldnt possibly do
  better without having negative distances,
  violations of laws of geometry

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Wavefront Planners
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Trulla
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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

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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,

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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

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Trulla Example
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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 thats it worth the
  computation?

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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