Page 285 - Introduction to Autonomous Mobile Robots

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 2 Chapter 6
1

1
 1 ---k  ----------- – ----- if ρ q() ≤ ρ
U rep q () =  2 rep  ρ q() ρ  0 (6.9)
0

 0 if ρ q() ≥ ρ 0
where k rep is again a scaling factor, ρ q() is the minimal distance from q to the object and
ρ the distance of influence of the object. The repulsive potential function U is positive
0 rep
q
or zero and tends to infinity as gets closer to the object.
If the object boundary is convex and piecewise differentiable, ρ q() is differentiable
everywhere in the free configuration space. This leads to the repulsive force F :
rep

F rep q () = – ∇ U rep q () (6.10)

 q – q

1
1
1
 k  ----------- – ----- ----------------------------------------- if ρ q() ≤ ρ
obstacle
F q () =  rep  ρ q() ρ  2 ρ q() 0
0 ρ q()
rep

 0 if ρ q() ≥ ρ 0
q
The resulting force F q() = F () + F q () acting on a point robot exposed to the
att rep
attractive and repulsive forces moves the robot away from the obstacles and toward the goal
(see figure 6.5). Under ideal conditions, by setting the robot’s velocity vector proportional
to the field force vector, the robot can be smoothly guided toward the goal, similar to a ball
rolling around obstacles and down a hill.
However, there are some limitations with this approach. One is local minima that appear
dependent on the obstacle shape and size. Another problem might appear if the objects are
concave. This might lead to a situation for which several minimal distances ρ q() exist,
resulting in oscillation between the two closest points to the object, which could obviously
sacrifice completeness. For more detailed analyses of potential field characteristics, refer
to [21].

The extended potential field method. Khatib and Chatila proposed the extended poten-
tial field approach [84]. Like all potential field methods this approach makes use of attrac-
tive and repulsive forces that originate from an artificial potential field. However, two
additions to the basic potential field are made: the rotation potential field and the task
potential field.
The rotation potential field assumes that the repulsive force is a function of the distance
from the obstacle and the orientation of the robot relative to the obstacle. This is done using