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

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Chapter 10: Metric Path Planning a. Representations b. Algorithms – PowerPoint PPT presentation

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


1
Chapter 10 Metric Path Planning a.
Representations b. Algorithms
2
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

3
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

4
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
5
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

6
Meadow Maps
  • Example of the basic procedure of transforming
    world space to cspace
  • Step 1 (optional) grow obstacles as big as robot

7
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

8
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?

9
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

11
Generalized Voronoi Graphs
  • Create lines equidistant from objects and walls,
  • Intersections of lines are nodes
  • Result is a relational graph

12
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?

13
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

14
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

15
Algorithms
  • Path planning
  • A for relational graphs
  • Wavefront for operating directly on regular grids
  • Interleaving Path Planning and Execution

16
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

17
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)
18
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)

19
Example A to E
1
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
20
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

21
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

22
Wavefront Planners
23
Trulla
24
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

25
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,

26
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

27
Trulla Example
28
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?

29
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
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