1. Dave Lattanzi’s Laplace Planner

2. Laplace Solution Store node charges as a dictionary – Originally used state variables and nodeList Find Laplace solution and store in new dictionary – Not a continuously updated field

3. Path Finding Search for neighbor with minimum charge difference Store as a dictionary of previous nodes

4. Results Numerical issues – What’s a good initial charge value? – What’s a good tolerance Gets stuck at local minima (shouldn’t happen) Can find path if initialized properly

5. LaPlace Implementation Ryan Keedy 5/4/2012

6. Algorithm Average surrounding nodes – If using 4 nodes, measure distance, and only average adjacent node values one unit away – Average in 1.0 for each node short of expected – Store values in “state” variable – Color code the map according to values Continue until the start node has been assigned a new value (non-initialized) Create the path without complicated math – Simply work from the start, progressing along neighbors with the smallest calculated values

7. Decisions to be Made How to initialize nodes – Any value below 1.0 will promote local minima – “Convergence” takes longer 4 vs. 8 nodes – 8 nodes can propagate calculations faster along diagonals By picking an arbitrary cutoff value, there are two possible bad outcomes – Everything was initialized at 1.0, and the averaging process hasn’t propagated to the start node – Other initialization value may result in local minima One problem with stopping when the start node is reached is that path may not be as smooth or “optimized”

8. Liang-Ting Jiang

9. # Identify boundary and free nodes def FindBoundary(self): # Visualize the Laplacian values as a grey map def GreyMap(self, drawList): # Greedily follow the lowest value def FollowGradient(self, start, goal): Laplace Path Planner # Update value: mean of 8 neighbors def computeLaplacian(self): Liang-TIng Jiang Initial value: Boundary: 1.0 Free: 0.5 Goal: 0.0

10. Laplace Planner – A Lewis Assumes nodes neighboring walls are walls themselves 500+ iterations of non-parallel computing Walls – 10000 Empty - 5000

11. Issues/Solutions? Occasionally gets stuck Try to minimize data thrown around Change area of interest – grow from goal?

12. MatLab Solution a=imread('LittleMap.jpg','jpg');%Imports Image imshow(a) %Displaces map to be processed figure(1) map=rgb2gray(a); %Creates Matrix Map to be manipulated figure(2) Simple image to start with, Black=Wall White=Goal Gray= Area for paths

13. imshow(map) i=2; j=2; [I,J]=size(map) % DIFFERENTIATION 1 while (i<I) while (j<J); Mapnew(i,j)=(map(i-1,j)+map(i+1,j)+map(i,j-1)+map(i,j+1))/4; j=j+1; end i=i+1; j=2; end i=2; j=2; %DIFFERENTIATION 2 while (i<I-1) while (j<J-1); Mapnew1(i,j)=(Mapnew(i-1,j)+Mapnew(i+1,j)+Mapnew(i,j-1)+Mapnew(i,j+1))/4; j=j+1; end i=i+1; j=2; end

14. Current Results Contours plot shows great results around edges. Needs to continue to slop toward goal!!!