16-735 Paper Presentation “Numerical Potential Field Techniques for Robot Path Planning” †
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Transcript of 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 config-uration using heuristic functions (e.g. tangent bug, po-tential field)
How to generate collision free paths?
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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)
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1. Incrementally build a graph connecting the local min-ima of potential functions defined over the configu-ration 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)
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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 free
is discretized in a 2D or 3D - dimensional grid and free:
- 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 free neighbor of boundary or of i to 1; than the neighbors of this new points to 2 and so on until all free has been explored;
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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
x
1- neighborhood (2D)
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Multiscale pyramid- 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)
0 W Wd{ ( ) } freex BM x
g
s
g
g
s
g
s
Robotic Motion Planning 16-735 Potential Field Techniques
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Basic functions3. Configuration space:
Q is also discretized in a n-dimensional grid GQ and GQfree
- the resolution is defined as the logarithm of the inverse of the distance between two discretizaton points - the resolution r of GW is also:
- for any given workspace resolution r, the corrisponding resolution Ri of the discretization of Q 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 controlpoints in 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
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↦W: ( )i ip free pV x V x ( ) ( ( ( , )), , ( ( , )))
i sp i p sU q G V X p q V X p q
Image from “Numerical Potential Field Techniques for Robot Path Planning”
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”
Image from “Numerical Potential Field Techniques for Robot Path Planning”
Image from “Numerical Potential Field Techniques for Robot Path Planning” 9
Robotic Motion Planning 16-735 Potential Field Techniques
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
- avoid zero value when one point have reached the goali
s
ii
s
is yyyyG111 maxmin),,(
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1.Best First motion
2.Random motion
3.Valley-guided motion
4.Constrained motion
Techniques to construct local-minima graph
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Valley Guided Motion Technique
Images from “Google Map 2007”
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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 qs 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:
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s
iiipiheur
s
ii
qpXV
qpXVqU
eqUipi
,)(
log)(),(
where is a small number
qstart
qgoal
qs
qg
V
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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 val-ley 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
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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 “con-
strained” 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
Best First Motion and Random Motion Technique
1. Best-First Motion Technique
2. Random Motion Technique
Agitation16
Robotic Motion Planning 16-735 Potential Field Techniques
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.
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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
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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 be-tween the different processing unit Random motion
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Random Motion Technique
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
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Random Motion Technique
PDF for Brownian Motion can be described as Gaussian Distribution Function
2
21( ) exp 22
ii
ii
qP qtt
0i
m
t
where duration of random motion
: resolution of elementi l
t N tq
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Probability of location of qi after time t (end of random motion)
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
0( )q t0( )q t t
0( )q t0( )q t t
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Random Motion Technique
Duration of Random Motion
Attraction Radius ( )( )iR locA q
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Mean Distance
1
2 221 1 1 1
2 2
Random walks for each step : where complex phaser of th step
( )
:
( ) ( ( )) ( )( )
k
k
k l k l
ji
k
Nj
ik
N N N Nj j j
i i i ik l k l
i i
ej k
z e
E z E e e E N e
E N N
iR iA t
No correlation between potential function and attraction radius Only probabilistic result can be obtained.
Random Motion Technique
Duration of Random Motion
Since attraction radius can’t exceed workspace diameter, by normalizing it to 1, we can obtain,
[1,d]
Diameter of 1
Recall that sup
Jacobian map from Configuration space to Workspace
i i
i isup R R sup
i isup
p R,q Q,j iisup i
J A A / J
xJ p,qq
J
W
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By taking the supremum of the Jacobian matrix, we can guarantee more coarse motion in workspace.
Random Motion Technique
Duration of Random Motion
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Finally, we have
21t
2
1iR
[ , ] i
A( ) max Dloc i nt q
Due to
Since attraction radius is strongly positive, we treat it as random variable with truncated Laplace Distribution of density.
Double exponential shape of Laplace Distribution of density
sup sup( ) exp( )i i
i iR Rp A J J A
At first we actually assume attraction radius is constant which is the same as diameter of workspace. However, if we have distribution function of attraction radius we also can have corresponding probability of t.
2( ) exp( )p t tt
21( ) /E t Expectation
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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