1. Sept, 19, 2007 NSH 3211 Hyun Soo Park, Iacopo Gentilini
Paper Presentation “Numerical 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 Cybernetics Volume 22, Issue 2, Mar/Apr 1992 , pages:
Robotic Motion Planning Potential Field Techniques

2. 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 robot’s configuration using heuristic functions (e.g. tangent bug, potential field)
Robotic Motion Planning Potential Field Techniques

3. Robotic Motion Planning 16-735 Potential Field Techniques
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 robot’s degrees of freedom)
2. Local approach:
advantages: no precomputation needed
disadvantages: - “search graph” considerably larger than “connectivity graph” dead ends (local minima)
Robotic Motion Planning Potential Field Techniques

4. Robotic Motion Planning 16-735 Potential Field Techniques
How to combine advantages of both?
Incrementally build a graph connecting the local minima of potential functions defined over the configuration space  (no expensive precomputation)
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)
Robotic Motion Planning Potential Field Techniques

5. Robotic Motion Planning 16-735 Potential Field Techniques
Basic functions
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
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 free
 is discretized in a n-dimensional grid  and  free:
- a 1-neighborhood is used, that means neighbors in 2D, 6 neighbors in 3D, and n 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; x
1- neighborhood (2D) 5
Robotic Motion Planning Potential Field Techniques

6. Robotic Motion Planning 16-735 Potential Field Techniques
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) g g  s s g g  s
Robotic Motion Planning Potential Field Techniques

7. Robotic Motion Planning 16-735 Potential Field Techniques
Basic functions
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:
Robotic Motion Planning Potential Field Techniques 7

8. Robotic Motion Planning 16-735 Potential Field Techniques
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 minima we can not assume that U is free
of local minima: it depends on the definition of G
where pi are the control points 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 (needed precomputation)
Robotic Motion Planning Potential Field Techniques

9. Robotic Motion Planning 16-735 Potential Field Techniques
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”
Robotic Motion Planning Potential Field Techniques

10. 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!):
- concurrent attraction causes local minima
- concurrent attraction compensed - avoid zero value when one point have reached the goal
Robotic Motion Planning Potential Field Techniques

11. Techniques to construct local-minima graph
Best First motion
Random motion
Valley-guided motion
Constrained motion
Robotic Motion Planning Potential Field Techniques

12. Robotic Motion Planning 16-735 Potential Field Techniques
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 qs and qg. At every crossroad a decision is made using an heuristic function defined as Q-potential Uheur
if step b. is successful, path is calculated, otherwise failure
qstart qs qg V
Best experimental Q-potential function:
qgoal
where s is a small number
Robotic Motion Planning Potential Field Techniques 12

13. Robotic Motion Planning 16-735 Potential Field Techniques
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
Robotic Motion Planning Potential Field Techniques 13

14. Robotic Motion Planning 16-735 Potential Field Techniques
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 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 the hyperplane 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:
Robotic Motion Planning Potential Field Techniques 14

15. Robotic Motion Planning 16-735 Potential Field Techniques
Best First Motion and Random Motion Technique
1. Best-First Motion Technique
2. Random Motion Technique
Agitation
Robotic Motion Planning Potential Field Techniques

16. Robotic Motion Planning 16-735 Potential Field Techniques
Best First Motion and Random Motion Technique
1. Best-First 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
2. Random Motion Technique
- The number of iteration can be specified by user so that this algorithm performs fast.
Robotic Motion Planning Potential Field Techniques

17. Robotic Motion Planning 16-735 Potential Field Techniques
Random Motion Technique
Local Minimum Detection
Limited number of searching iteration
If U(q) > U(q’), then q’ is successor
 Gradient motion
If NO q’, then q is local minimum
Robotic Motion Planning Potential Field Techniques

18. Robotic Motion Planning 16-735 Potential Field Techniques
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
Robotic Motion Planning Potential Field Techniques

19. Robotic Motion Planning 16-735 Potential Field Techniques
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.
Drawback : No guarantee to find a path whenever one exists.
However, by property of Brownian Motion, as the number of iteration of random motion,
Robotic Motion Planning Potential Field Techniques

20. Robotic Motion Planning 16-735 Potential Field Techniques
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)
Robotic Motion Planning Potential Field Techniques

21. Robotic Motion Planning 16-735 Potential Field Techniques
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 ( )
Robotic Motion Planning Potential Field Techniques

22. Robotic Motion Planning 16-735 Potential Field Techniques
Random Motion Technique
Duration of Random Motion
Attraction Radius ( )
No correlation between potential function and attraction radius
 Only probabilistic result can be obtained.
Robotic Motion Planning Potential Field Techniques

23. Robotic Motion Planning 16-735 Potential Field Techniques
Random Motion Technique
Duration of Random Motion
Since attraction radius can’t exceed workspace diameter, by normalizing it to 1, we can obtain,
By taking the supremum of the Jacobian matrix, we can guarantee more coarse motion in workspace.
Robotic Motion Planning Potential Field Techniques

24. Robotic Motion Planning 16-735 Potential Field Techniques
Random Motion Technique
Duration of Random Motion
Since attraction radius is strongly positive, we treat it as random variable with truncated Laplace Distribution of density.
Finally, we have
Double exponential shape of
Laplace Distribution of density
Due to
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.
Expectation
Robotic Motion Planning Potential Field Techniques

25. Robotic Motion Planning 16-735 Potential Field Techniques
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
Robotic Motion Planning Potential Field Techniques