Step by Step Construction of the Formula


Complete Formula

   The complete formula is the following, and the variables are explained below:

  Theta = heading direction
  Psi_G = direction of goal (in fixed, exocentric frame of reference)
  d_g   = distance to goal
  Psi_O = direction of obstacle (in fixed, exocentric frame of reference)
  d_o   = distance to obstacle

  V              = Speed
  B              = Damping
  K, G1, G2      = Goal parameters
  E1, E2, O1, O2 = Obstacle parameters

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Construction of the Damping Part
 

 
  The followings are example graphs demonstrating the contribution of the damping part to the computation of rotation acceleration with B = 3.25
 
 
B = damping constant
Damping values are plotted against the "angle to goal" (goal direction with respect to the agent's heading) values. Zero plane is also shown explicitly in green.

 

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Construction of the Target Part
 

 
  The following graphs are plotted using these example values:

  Until now, we prepared the "decreasing influence" of the goal, now we'll also include the effect of goal direction. Below are 4 different views of the resulting graph:
 

  To see the effect of heading on the target part, let's plot the target contribution against heading (with psi_G = 60 degrees) instead of the net angle to goal. The result is simply a shift (equal to psi_G) of the graph values. Below are 4 views of the resulting graph (with the same corresponding view angles as above graphs):
 

 

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Construction of the Obstacle Part
 

 
  The following values are used while drawing the graphs:
 

  The first section of the formula which deals with the effect of distance is ready. The second section dealing with the magnitude of the "angle to obstacle" is constructed below:
 
Three different views for the first step are provided on the right and on the following row:
Three different views for the second step are provided on the right and on the following row:
Three different views for the third step are provided on the right and on the following row:

  Now we combine the "magnitude of the angle" influence and the distance influence  to get the following graph (shown in 4 different views):
 

  Last step is to include the obstacle direction influence. Below is the final graph showing the effect of a single obstacle (in 4 different views):
 

 

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