1. Final Project Presentation on autonomous mobile robot
Submitted to
Prof, Jaebyung Park
Robotics Lab
Submitted by
Ansu Man Singh
Student ID ( )

2. Outline Title Objective Procedure Binary Image
Attractive Potential field
Repulsive Potential field
Total field
Gradient descent
Navigation function

3. Title
Path planning using attractive and Reflective potential field

4. Objective
To generate a path for mobile robot using attractive and repulsive potential field

5. Procedure Goal Position Attractive Potential field Binary Image
Repulsive Potential field
Start Position
Gradient Descent Algorithm
Required Path

6. Binary Image Binary Image of 200 by 200 pixel is taken
1’ will represent free space, 0’s will represent obstacle space
example
Obstacles

7. Attractive Potential Field
The Attractive Potential field is generated using conic and quadratic functions
Quadratic
Conic

8. Attractive Potential Field
Attractive Potential function
Goal Position

9. Attractive Potential Field
Code Section
goal_pos = [ ];
Uatt = zeros(wSpace_Size);
d_xtrix_goal =3;
K=0.06;
const1 = 0.5*K*d_xtrix_goal^2;
for i=1:wSpace_Size(1)
for j=1:wSpace_Size(2);
A=(goal_pos(1)-i)^2+(goal_pos(2)-j)^2;
distance=sqrt(A);
if(distance > d_xtrix_goal)
Uatt(i,j)=d_xtrix_goal*K*distance -const1;
else
Uatt(i,j)=0.5*d_xtrix_goal*K*distance^2 ;
end

10. Repulsive Potential Field
Repulsive function used
Repulsive function is generated by the help of binary image.
Steps used in generating Repulsive function
Find the obstacle position in the binary image
Generate field using the equation for the distance Q* above and below the obstacle pixel position

11. Repulsive Potential Field

12. Repulsive Potential Field
Code section
for i=1:wSpace_Size(1)
for j=1:wSpace_Size(2);
if(wSpace_Bin(i,j)==0)
Uref(i,j)=8;
for k= -xtrix_OBS:xtrix_OBS
for p =-xtrix_OBS:xtrix_OBS
if((i+k)>wSpace_Size(1)||(i+k)<1||(j+p)>wSpace_Size(2)||(j+p)<1)
else
if(wSpace_Bin(i+k,j+p)~= 0)
distance2 = sqrt((k)^2+(p)^2);
Uref(i+k,j+p)=Uref(i+k,j+p)+ 0.5*2*(1/distance2 - 1/xtrix_OBS)^2;
Uref(i+k,j+p)= 8;
end
Uref(i,j) = Uref(i,j) +0;

13. Total Field
Total Potential field is addition of attractive and Repulsive field +

14. Gradient Descent Algorithms used to find the path in the field
Gradient descent always follows negative slope
Input: A means to compute the gradient ∇U (q)at a point q
Output: A sequence of points {q(0), q(1), ..., q(i)}
1: q(0) = qstart
2: i = 0
3: while ∇U (q(i)) ≠= 0 do
4: q(i + 1) = q(i) + α(i)∇U (q(i))
5: i = i + 1 6: end while

15. Gradient Descent Code section while(flag)
position(iteration+1,:) = [pos_x pos_y U_tot(pos_y,pos_x)];
pos_x = ceil(pos_x+ alpha*grad_x);
pos_y = ceil(pos_y+ alpha*grad_y);
if((grad_x==0&&grad_y==0)||iteration >1000)
flag = 0;
end
if (pos_x>=200||pos_y>=200)
flag =0;
else
grad_x=-fx(pos_y,pos_x);
grad_y=-fy(pos_y,pos_x);
iteration= iteration+1;

16. Gradient Descent
Contour map of field with path
Start point = (0,80)

17. Gradient Descent
Path 2

18. Gradient Descent
Local Minima problem

19. Gradient Descent Local Minima problem can be using navigation function
Navigation function definition
A function is called a navigation function if it
is smooth (or at least Ck for k ≥ 2),
has a unique minimum at qgoal in the connected component of the free space that contains qgoal,
is uniformly maximal on the boundary of the free space, and
is Morse.

20. Navigation function Navigation function on sphere world
Obstacle functions
Distance to goal function

21. Navigation function
Switch function which is used to map from (0 to infinity) to [0 1]
Sharpening function to make critical points non-degenerate

22. Navigation function
Final navigation function on sphere world

23. Navigation function
Implementation of navigation function on sphere world

24. Navigation function Code section clear all ; x= -10:0.1:10;
y= -10:0.1:10;
x_goal = 8;
y_goal = 8;
K= 4;
nav_fxn = zeros(length(x),length(y));
lambda = 2;
for i = 1 :length(x)
for j = 1:length(y)
beta = beta_function(x(i),y(j));
dist_goal = norm([x(i)-x_goal y(j)-y_goal],2);
radius = norm([x(i) y(j)],2);
if(radius>10)
nav_fxn(i,j) = 1;
else
nav_fxn(i,j) = dist_goal^2/(dist_goal^(2*K) + lambda*abs(beta))^(1/K);
end

25. Navigation function
Conversion from star-shaped set to sphere shaped set
This conversion is essential for representation of object in real world.

26. References
[1] Howie Choset et al, Principle of robot Motion-Theory, Algorithms and Implementation,
[2]Elon Rimon, Daniel E Koditschek, Exact Robot Navigation Using Artificial Potential Functions

27. Thank You