Step by Step Construction of the Formula



Complete Formula

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


 

  U_art = The artificial potential field, which the agent is subjected to
  U_xd  = Attractive potential field created by the goal
  U_O   = Repulsive potential field created by the obstacle
  x     = position of agent
  x_d   = goal position
  k_p   = position gain
  eta   = constant gain
  rho   = shortest distance to the obstacle
  rho_0 = limit distance of the potential field influence
 
 

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Construction of the Attractive Potential Field
 

 
  The followings are example graphs demonstrating the attractive potential field computed for k_p = 1. The fourth 2-D graph summarizes the computation.
 
 

 

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Construction of the Repulsive Potential Field
 

 
  The following graphs are plotted using the values below, and also note that the tip of the cones are actually extending to infinity but they are cut for demonstration purposes. The last two 2-D graphs summarize the computation:

 

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