16-735 Paper Presentation

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16-735 Paper PresentationNumerical Potential
Field Techniques for Robot Path Planning

• Sept, 19, 2007
• NSH 3211
• Hyun Soo Park, Iacopo Gentilini

Barraquand, J., Langlois, B., and Latombe,
J.-C.IEEE Transactions on Systems, Man and
CyberneticsVolume 22, Issue 2, Mar/Apr 1992 ,
pages 224 - 241
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How to generate collision free paths?
1. Global approach

• building a connectivity graph of collision
  free configuration
• searching the graph for a path (e.g. network
  of one dimensional curves)

Image from Numerical Potential Field Techniques
for Robot Path Planning
2. Local approach
- searching a grid placed across the robots
configuration using heuristic functions (e.g.
tangent bug, potential field)
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Differences between global and local?
1. Global approach

• advantages very quick search in the
  connectivity graph
• disadvantages expensive precomputation step to
  get the graph (exponential in the dimension n
  of the configuration space Q where n is number
  of the robots degrees of freedom)

2. Local approach

• advantages no precomputation needed
• disadvantages - search graph considerably
  larger than
  connectivity graph
  - dead ends (local minima)

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How to combine advantages of both?

1. Incrementally build a graph connecting the local
   minima of potential functions defined over the
   configuration space ? (no expensive
   precomputation)
2. Concurrently searching this graph until the goal
   is reached ? escaping local minima (search within
   much smaller search graph)

• Based on multiscale pyramids of bitmap arrays of
  ? and ? (not analytically defined potential
  function)

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Basic functions
1. Forward kinematic X R Q ? W (p,
q) ? x X (p, q) where p ? R is a point
in the robot

• Workspace bitmap BM W ? (1,0)
• x ? BM (x) where BM(x)
  0 represents Wfree
• - discrete grid GW workspace
  representation is given as a grid at a 512512
  level of resolution using a scaling factor 2 a
  pyramid of representations is also computed
  until the coarsest resolution level 1616 is
  reached - ? is the distance between two
  adjacent points (? min 1/512 and ? max
  1/16 if given in percentage of the workspace
  diameter)
• - a 1-neighborhood is used, that means 4
  neighbors in 2D, 6 neighbors in 3D, and 2n
  neighbors in n-D within the discrete grid
• - preparation a wavefront expansion is
  computed by setting each point in GWfree
  neighbor of boundary or of GWOi to 1 than the
  neighbors of this new points to 2 and so on
  until all GWfree has been explored

• k-neighborhood with k ? 1,r of a point x in a
  grid of dimension r is defined as the set of
  points in the grid having at most k coordinates
  differing from those of x
• k 1 ? 2 r points
• k 2 ? 2 r2 points
• k r ? 3r -1points

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Basic functions

• Configuration space? is also discretized in a
  n-dimensional grid ?? and ?? free
• - the resolution is defined as the
  logarithm of the inverse of the distance
  between two discretizaton points - the
  resolution r of ?? is also
• - for any given workspace resolution r,
  the corrisponding resolution Ri of the
  discretization of ? along the qi axis is chosen
  in such a way that a modification of qi by
  ?i generates a small motion of the robot (any
  point p of R moves less than nbtol ? )
• where

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How are potential functions built?
W-potential - computed in W
Q-potential - defined over Q
where G is called the arbitration
function - good Q-potential in ?
(whose dimension is big) - if Vpi are
free of local minimawe can not assume that U is
free of local minima it depends on
thedefinition of G
where pi are the control pointsin the
robot R - small dimension of ? (2 or 3)
for low cost information - have to be
built such that they are free of local minima
(neededprecomputation)
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W-Potential
1. Simple W-Potential

• get the position of control point p in ? and its
  goal position xgoal
• set Vp 0 at xgoal
• set the neighbors in ?? free of xgoal to 1 and
  so on

Image from Numerical Potential Field Techniques
for Robot Path Planning
-?Vp is the direction to goal
2. Improved W-Potential

• build the workspace skeleton S as subset of ??
  free computing the wavefront expansion
• connect xgoal to ? and compute Vp in the
  augmented S using a queue of points of S sorted
  by decreasing value
• compute Vp in ?? free \ ? as shown in 1.

Image from Numerical Potential Field Techniques
for Robot Path Planning
Image from Numerical Potential Field Techniques
for Robot Path Planning
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Q-Potential

• attracts control points pi toward their
  respective goal position
• arbitration function definition (minimize local
  minima!)

• - concurrent attraction causes local minima

• - concurrent attraction compensed- avoid zero
  value when one point have reached the goal

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Techniques to construct local-minima graph

1. Best First motion
2. Random motion
3. Valley-guided motion
4. Constrained motion

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Best First Motion and Random Motion Technique
1. Best-First Motion Technique
2. Random Motion Technique
Agitation
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Best First Motion and Random Motion Technique
1. Best-First Motion Technique

• - Good for n lt 4
• What if n is getting bigger?
• ? Searching unit increases in almost exponential
  order ( ) as increasing DOF
• ? Thus, we need another algorithm to search
  local minima

2. Random Motion Technique
- The number of iteration can be specified by
user so that this algorithm performs fast.
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Random Motion Technique
Local Minimum Detection
Limited number of searching iteration If U(q) gt
U(q), then q is successor ? Gradient motion If
NO q, then q is local minimum
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Random Motion Technique
Path Joining Adjacent Local Minima
Smoothing
This can be performed concurrently on a parallel
computer because of no need to communicate
between the different processing unit ? Random
motion
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Random Motion Technique
Dead-end
No more local minima near current position
Drawback No guarantee to find a path whenever
one exists. However, by property of Brownian
Motion, as the number of iteration of random
motion,
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Random Motion Technique
PDF for Brownian Motion can be described as
Gaussian Distribution Function
Probability of location of qi after time t (end
of random motion)
At the boundary of obstacles, usually random
motion reflects to tangent hyperplane of
obstacles when motion collides against obstacles
but this paper implemented as substituting by
new random motion generation.
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Random Motion Technique
Duration of Random Motion
Should not be too short ? No chance to escape
Should not be too long ? waste of time and no
gradient motion
Attraction Radius ( )
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Random Motion Technique
Duration of Random Motion
Since attraction radius cant exceed workspace
diameter, by normalizing it to 1, we can obtain,
Finally, we have
Due to
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Valley Guided Motion Technique
Searching valleys V of Q-potential U in Qfree

• using -?U calculated in qstart and qgoal reach to
  local minima qi and qg
• search V for a path connecting qi and qg.
  Atevery crossroad a decision is made using
  anheuristic function defined as Q-potential
  Uheur
• if step b. is successful, path is
  calculated,otherwise failure

Best experimental Q-potential function
Image from Numerical Potential Field Techniques
for Robot Path Planning
where s is a small number
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Valley Guided Motion Technique
When a point q ? Q is a valley points (q ? V)?

• compute U(q)
• compute the 2n values of U at the 1-neihbors of
  q
• for each possible valley direction i ? 1,n
• compare U(q) to the 2n 2 values of U at the
  1-neighbors in the hyperplane orthogonal to the
  qi axis
• if U(q) is smaller or equal to these 2n 2
  values, q is a valley point.

q
n 2
- complexity is O(n2) or if using 2-neighborhood
O(n4) - better using n-neighborhood but
cardinals are 3n-1 with exponential complexity
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Constrained Motion Technique
Starting from qstart in Qfree

• follow -?U flow until local minima qloc is
  attained
• if qloc qgoal the problem is solved otherwise
  execute a constrained motion Mi(qloc) or
  -Mi(qloc) with i ? 1,n
• increase iteratively the i th configuration space
  coordinate by the increment ?i until a saddle
  point of the local minimum well is reached (U
  decreases again). If (q1,, qi, ,qn) is the
  current configuration its successor minimizes U
  over the set consisting of (q1,, qi ?i ,,qn)
  and its 1-neighbors in thehyperplane orthogonal
  to the qi axis (the motion thus track a valley in
  the (n-1)-dimensional subspace orthogonal to the
  qi axis).
• terminate the constrained motion and execute an
  other gradient motion

qloc
n 2
Q-potential function used
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Conclusion
Approach - Constructing a potential field over
the robots configuration - Building a
graph connecting the local minima of the
potential - Searching graph Aim Escaping
local minima 4 techniques - Best-first
motion gives excellent result with few DOF
robots (n lt 5) - Random motion gives good
results with many DOF - Valley-Guided motion
inferior result but can be improved in future -
Constrainted motion good at planning the
coordinated motions