Final Project Presentation on autonomous mobile robot

Submitted toProf, Jaebyung Park

Robotics LabSubmitted by

Ansu Man SinghStudent ID (201150875)

Outline

• Title• Objective• Procedure• Binary Image• Attractive Potential field• Repulsive Potential field• Total field• Gradient descent• Navigation function

Title

• Path planning using attractive and Reflective potential field

Objective

• To generate a path for mobile robot using attractive and repulsive potential field

Procedure

Binary Image

Attractive Potential field

Repulsive Potential field

Gradient Descent Algorithm

Required Path

Goal Position

Start Position

Binary Image

– Binary Image of 200 by 200 pixel is taken– 1’ will represent free space, 0’s will represent

obstacle space– example

Obstacles

Attractive Potential Field

• The Attractive Potential field is generated using conic and quadratic functions

QuadraticConic

Attractive Potential Field

• Attractive Potential function

Goal Position

Attractive Potential Field

• Code Sectiongoal_pos = [180 180];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 endend

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

Repulsive Potential Field

• Repulsive Potential Field

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; else Uref(i+k,j+p)= 8; end end end end else Uref(i,j) = Uref(i,j) +0; end endend

Total Field• Total Potential field is addition of attractive

and Repulsive field

+

Gradient Descent• Algorithms used to find the path in the field• Gradient descent always follows negative

slopeInput: 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

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; end end

Gradient Descent• Contour map of field with path

Start point = (0,80)

Gradient Descent• Path 2

Gradient Descent• Local Minima problem

Gradient Descent• Local Minima problem can be using

navigation function• Navigation function definition

A function is called a navigation function if it1.is smooth (or at least Ck for k ≥ 2),2.has a unique minimum at qgoal in the connected component of the free space that contains qgoal,3.is uniformly maximal on the boundary of the free space, and4.is Morse.

Navigation function • Navigation function on sphere world• Obstacle functions

• Distance to goal function

Navigation function • Switch function which is used to map from (0

to infinity) to [0 1]

• Sharpening function to make critical points non-degenerate

Navigation function • Final navigation function on sphere world

Navigation function • Implementation of navigation function on

sphere world

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 endend

Navigation function

• Conversion from star-shaped set to sphere shaped set

• This conversion is essential for representation of object in real world.

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

Thank You