Evolution of a Brachystochrone

Evolving a Brachystochrone using Genetic Algorithms

evolution of a Brachystochrone - problem info

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 and its velocity is 0 at the startpoint.

chromosome (genotype representation)

The track is modelled by lines through successive trackpoints.

  // 10 trackpieces => 9 inner trackpoints
  double object_parameters[]= {10,10, 20,20, 30,30, 40,40, 50,50, 
                               60,60, 70,70, 80,80, 90,90, 100,100, 110,110 };
  double strategy_parameters[] = {0,0, 0,1, 0,1, 0,1, 0,1,     
                                  0,1, 0,1, 0,1, 0,1, 0,1, 0,0 }; 

 As we want to optimize the height of the tracks inner points all but the y-positions
 of the inner track points have a strategy parameter of 0.  
 

   // 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)

quality function

The time a moveable point P needs to slide frictionless on a track from a starting point to a destination driven by means of his own gravity should 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;