evolution of a Brachystochrone

The goal is to optimize the track between a start- and an endpoint in such a way, that the time a moveable point sliding down the track needs to reach the end of the track becomes minimal. The point moves frictionless by means of his own gravity.

evolving a Brachystochrone using genetic algorithms

GA - interface help

Genetic Algorithms Theory should be an applet here

Evolution of a Brachystochrone - Problem Info

chromosome (genotype representation)

genetic algorithms
Only the y-positions of the inner tack points are encoded in the chromsome.

   // 10 trackpieces => 9 inner trackpoints => 9*10 bit gray-encoded bit-vector
   [ [010011110],[011000110],[110100001],[010111011],[101111001], 
     [101011001],[010011101],[011000011],[001111001] ]

drawing the chromsome (phenotype representation)

evolution strategy and gentic algorithms
draw a polygon through track-points

quality function

The time a moveable point P needs, to slide frictionless on a track from a startpoint to an endpoint, driven by means of his own gravity shall become minimal, by optimizing the track between start and end.

t_i is the time needed to slide down the track from (x_i,y_i) to (x_i+1,y_i+1);
t_i is calculated by integrating the acceleration:

If the point on track-piece i is accelerated by a_i we get its position, after being accelerated for a time t, if we have its initial velocity v_i and its initial position x_i.

By inserting a_i and l_i,j we get:

and we solve this for t, to get t_i;

applet parameters

 
    param name="pieces"  value="10" 

       the track will be divided into 10 trackpieces
  
   param name="y_start" value="10" 

       height of the track at start position (0 highest !)
  
   param name="y_end"   value="30" 

       height of the track at end position (120 lowest !)
  
   param name="x_inc"   value="10" 

       horizontal distance between track points