Motion Planning for a Point Robot

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Motion Planning for a Point Robot
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Problem
free space
obstacle
free path
obstacle
obstacle
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Bug Algorithms

• Assumptions
• The robot is modeled as a point
• The obstacles have bounded perimeters and are in
  finite number
• The robot senses perfectly its position and can
  measure traveled distance
• The robot can perfectly detect contacts and their
  orientations
• The robot can compute the direction to the goal
  and the distance between two points, and has
  small amount of memory

Finish
Start
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Bug-0 Algorithm

• Bug-0
• Repeat
• Head toward the goal
• If the goal is attained then stop
• If contact is made with an obstacle then follow
  the obstacles boundary (toward the left) until
  heading toward the goal is possible again.

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Is Bug-0 Guaranteed to Work?
No!
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Bug-1 Algorithm

• Bug-1
• Repeat
• Head toward the goal
• If the goal is attained then stop
• If contact is made with an obstacle then
  circumnavigate the obstacle (by wall-following),
  remember the closest point Li to the goal, and
  return to this point by the shortest
  wall-following path

L2
Finish
L1
Start
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Distance Traveled T by Bug-1?

• Lower bound?
• T ? D (where D is the straight-line distance
  from Start to Finish)
• Upper bound?
• T ? D 1.5?SPi
• (where SPi is the sum of the perimeters of all
  the obstacles)

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Distance Traveled T by Bug-1?

• Lower bound?
• T ? D (where D is the straight-line distance
  from Start to Finish)
• Upper bound?
• T ? D 1.5?SPi
• (where SPi is the sum of the perimeters of all
  the obstacles)

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Distance Traveled T by Bug-1?

• Lower bound?
• T ? D (where D is the straight-line distance
  from Start to Finish)
• Upper bound?
• T ? D 1.5?SPi
• (where SPi is the sum of the perimeters of all
  the obstacles)

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Bug-2 Algorithm

• Bug-2
• Repeat
• Head toward the goal along the goal-line
• If the goal is attained then stop
• If a hit point is reached then follow the
  obstacles boundary (toward the left) until the
  goal-line is crossed at a leave point closer to
  the goal than the previous hit point

Finish
leave point
hit point
goal-line
Start
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Path Followed by Bug-2?
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Distance Traveled T by Bug-2?

• Lower bound?
• T ? D (where D is the straight-line distance
  from Start to Finish)
• Upper bound?
• T ? D SniPi
• (where Pi is the perimeter of obstacle i, ni is
  the number of hit points in obstacle i, and the
  sum S is taken over all the obstacles)

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Distance Traveled T by Bug-2?

• Lower bound?
• T ? D (where D is the straight-line distance
  from Start to Finish)
• Upper bound?
• T ? D SniPi
• (where Pi is the perimeter of obstacle i, ni is
  the number of hit points in obstacle i, and the
  sum S is taken over all the obstacles)

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Which one --- Bug-1 or Bug-2 --- does better?
Finish
Start
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Planning requires models

• The Bug algorithms are reactive motion
  strategies they are not motion planners
• To plan its actions, a robot needs a (possibly
  imperfect) predictive model of its actions, so
  that it can choose among several possible courses
  of action

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Point Robot on a Grid

• Assumptions
• The robot perfectly controls its actions
• It has an accurate geometric model of the
  environment (i.e., the obstacles)

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Now, the robot can search its model for a
collision-free path to the goal
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Search Algorithm

• SEARCH(Start, Finish)
• INSERT(Start,FRINGE)
• Repeat
• If FRINGE is empty then return failure
• q ? REMOVE(FRINGE)
• For every new position q in SUCCESSORS(q)
• Install q as a child of q in the search tree
• If q Finish then return a path from Start to
  Finish
• INSERT(q,FRINGE)

Start
FRINGE
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Blind Search Strategies

• Breadth-firstNew positions are inserted at the
  end of FRINGE
• Depth-firstNew positions are inserted at the
  beginning of FRINGE

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1000?1000 grid ? 1,000,000 configurations In 3-D
? 109 configurations In 6-D ? 1018
configurations!!! ? Need for sparser
discretization or smart search
techniques
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Visibility Graph
SHAKEY (SRI, 1969)
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Smart Search

• SEARCH(Startt,Finish)
• INSERT(qstart,FRINGE)
• Repeat
• If FRINGE is empty then return failure
• q ? REMOVE(FRINGE)
• For every new configuration q in SUCCESSORS(q)
• Install q as a child of q in the search tree
• If q Finish then return a path from Start to
  Finish
• INSERT(q,FRINGE)

? Smart ordering of the configurations in
FRINGE (best-first search)
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Attractive/Repulsive Potential Fields
Equipotential contours
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Attractive/Repulsive Potential Fields
Best-first search with potential fields Sort
positions in FRINGE in increasing order of
potential
Bug motion strategy with potential fields
Equipotential contours
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Local-Minimum Problem
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Path Shortening

• Two approaches
• Sort positions in FRINGE using an evaluation
  function of the form g(q) aU(q)
• Iteratively shorten a path after one has been
  produced.

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Application to Animating a Digital Actor on Flat
Terrain
The actor The environment
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PrincipleBound the actor by a cylinder and
project all objects the ground
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Shrink the disc representing the actor to a point
and grow the obstacles accordingly
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Motion Capture
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Integration of Motion Planning and Motion Capture
High-Level Navigation Goals
Fast 2D Path Planner
Path
Path-Following Controller
Body Posture
Graphic Display
Base Point PD Controller
Motion Capture Data
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Simulated Vision
Actors view
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Perception-Based Planning
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Perception-Based Planning
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video
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video
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