16-735 paper presentation “numerical potential field techniques for robot path planning” †

16-735 Paper Presentation“Numerical Potential Field Techniques

for Robot Path Planning” †

Sept, 19, 2007NSH 3211

Hyun Soo Park, Iacopo Gentilini

Robotic Motion Planning 16-735 Potential Field Techniques

1

† Barraquand, J., Langlois, B., and Latombe, J.-C.IEEE Transactions on Systems, Man and CyberneticsVolume 22, Issue 2, Mar/Apr 1992 , pages: 224 - 241

1. Global approach:

2. Local approach:

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

- searching a grid placed across the robot’s configuration using heuristic functions (e.g. tangent bug, potential field)

How to generate collision free paths?

2Robotic Motion Planning 16-735 Potential Field Techniques

Image from “Numerical Potential Field Techniques for Robot Path Planning”

1. Global approach:

2. Local 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 robot’s degrees of freedom)

Differences between global and local?

- advantages: no precomputation needed

- disadvantages: - “search graph” considerably larger than “connectivity graph” - dead ends (local minima)


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”)

How to combine advantages of both?

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


Basic functions1. Forward kinematic:

X: R × Q → W (p, q) ↦ x = X (p, q) where p R is a point in the robot

2. 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 512×512 level of resolution; using a scaling factor 2 a pyramid of representations is also computed until the coarsest resolution level 16×16 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; 5Robotic Motion Planning 16-735 Potential Field Techniques

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

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

Robotic Motion Planning 16-735 Potential Field Techniques 6

Basic functions

3. 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 Δi generates a small motion of the robot (any point p of R moves less than nbtol × ) :

where:

How are potential functions built?

W-potential:- computed in W

Q-potential:- defined over Q

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)

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


↦W: ( )i ip free pV x V x ( ) ( ( ( , )), , ( ( , )))

i sp i p sU q G V X p q V X p q

W-Potential1. Simple W-Potential:

a. get the position of control point p in and its goal position xgoal

b. set Vp = 0 at xgoal

c. set the neighbors in free of xgoal to 1 and so on

-Vp is the direction to goal

2. Improved W-Potential:

a. build the workspace skeleton S as subset of free computing the “wavefront” expansion

b. connect xgoal to and compute Vp in the augmented S using a queue of points of S sorted by decreasing value

c. compute Vp in free \ as shown in 1.



Image from “Numerical Potential Field Techniques for Robot Path Planning” 8


Q-Potential

- attracts control points pi toward their respective goal position

- arbitration function definition (minimize local minima!):

s

iis yyyG

11 ),,( - concurrent attraction

causes local minima

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

i

s

ii

s

is yyyyG

111 maxmin),,(



1.Best First motion

2.Random motion

3.Valley-guided motion

4.Constrained motion

Techniques to construct local-minima graph

Best First Motion and Random Motion Technique

1. Best-First Motion Technique

2. Random Motion Technique

Agitation11


Best First Motion and Random Motion Technique

1. Best-First Motion Technique

2. Random Motion Technique

- Good for n <= 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

n3 -1

- The number of iteration can be specified by user so that this algorithm performs fast.


Random Motion Technique

Local Minimum Detection

q q

Limited number of searching iterationIf U(q) > U(q’), then q’ is successor

Gradient motionIf NO q’, then q is local minimum

q



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 14



Dead-end No more local minima near current position

Backtrack to arbitrary point in line of to which is selected by uniform distribution law.Then try to find another local minima.

initq locq

Drawback : No guarantee to find a path whenever one exists.

However, by property of Brownian Motion, as the number of iteration of random motion,

( ) 1goalP q



PDF for Brownian Motion can be described as Gaussian Distribution Function

2

2

1( ) exp

22i

iii

qP q

tt

0

i

m

t

where

duration of random motion

: resolution of elementi l

t N t

q

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. 16


Probability of location of qi after time t (end of random motion)


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

0( )q t

0( )q t t

0( )q t0( )q t t

Attraction Radius ( )( )iR locA q

2

[1, ]

1 gaurantees 68%

( ) max

i

i

R

Rloc i n

i

A

At q



Duration of Random Motion

Since attraction radius can’t exceed workspace diameter, by normalizing it to 1, we can obtain,

[1,d]

Diemeter of 1

Recall that sup

i i

i iR sup R sup

i isup

p R,q Q,j i

isup i

A J A / J

xJ p,q

q

J

W

Finally, we have

2

1t


2

1

iR

[ , ]i

A( ) max

Dloc i nt q

Due to

Valley Guided Motion Technique

Searching valleys V of Q-potential U in Qfree

a. using -U calculated in qstart and qgoal reach to local minima qi and qg

b. search V for a path connecting qi and qg. Atevery crossroad a decision is made using anheuristic function defined as Q-potential Uheur

c. if step b. is successful, path is calculated,otherwise failure


Best experimental Q-potential function:


s

iiipiheur

s

ii

qpXV

qpXVqU

eqUipi

,)(

log)(),(

where is a small number


Valley Guided Motion Technique

When a point q Q is a valley points (q V)?

1. compute U(q);2. compute the 2n values of U at the 1-neihbors of q;3. for each possible valley direction i [1,n]

a. compare U(q) to the 2n – 2 values of U at the 1-neighbors in the hyperplane orthogonal to the qi axis

b. if U(q) is smaller or equal to these 2n – 2 values, q is a valley point.

n = 2 q

- complexity is O(n2) or if using 2-neighborhood O(n4) - better using n-neighborhood but cardinals are 3n-1 with exponential complexity


Constrained Motion Technique

Starting from qstart in Qfree

1. follow -U flow until local minima qloc is attained;2. if qloc = qgoal the problem is solved; otherwise execute a

“constrained” motion +Mi(qloc) or -Mi(qloc) with i [1,n] :a. increase iteratively the i th configuration space coordinate

by the increment Δi 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+ Δi ,…,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).

b. terminate the constrained motion and execute an other gradient motion;

Q-potential function used:

s

iiipi qpXVqU ,)(

qloc

n = 2


22

Conclusion

Approach :- Constructing a potential field over the robot’s 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 <

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

16-735 paper presentation “numerical potential field techniques for robot path planning” †

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