Page 284 - Introduction to Autonomous Mobile Robots

Planning and Navigation

∂ U 269
-------
∇ U = x ∂ (6.2)
∂ U
-------
y ∂

The potential field acting on the robot is then computed as the sum of the attractive field
of the goal and the repulsive fields of the obstacles:

q
Uq() = U () + U q () (6.3)
att rep
Similarly, the forces can also be separated in a attracting and repulsing part:

q
F q() = F () F rep q ()
–
att
–
q
F q() = – ∇ U () ∇ U q () (6.4)
att rep
Attractive potential. An attractive potential can, for example, be defined as a parabolic
function.
1 2
U () = ---k att ⋅ ρ goal q () (6.5)
q
att
2
where k att is a positive scaling factor and ρ goal q () denotes the Euclidean distance
q – q . This attractive potential is differentiable, leading to the attractive force F
goal att
F () = – ∇ U () (6.6)
q
q
att
att
∇
q
F () = k – att ⋅ ρ goal q () ρ goal q () (6.7)
att
F () = k – ( ⋅ q – q ) (6.8)
q
att att goal
that converges linearly toward 0 as the robot reaches the goal.
Repulsive potential. The idea behind the repulsive potential is to generate a force away
from all known obstacles. This repulsive potential should be very strong when the robot is
close to the object, but should not influence its movement when the robot is far from the
object. One example of such a repulsive field is