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4 The Reactive Paradigm
an observer would see a robot behave in that ﬁeld: if the robot is far away
from the object, it will turn and move quickly towards it, then slow up to
keep from overshooting and hitting the object. Mathematically, this is called
LINEAR DROP OFF a linear drop off, since the rate at which the magnitude of the vectors drops off
can be plotted as a straight line. The formula for a straight line is y = mx + b,
where x is the distance and y is magnitude. b biases where the line starts,
and m is the slope (m = y ). Any value of m and b is acceptable. If it is not
x
speciﬁed, m = or -1 (a 45 slope up or down) and 1 b = in linear functions. 0
The linear proﬁle in Fig. 4.14b matches the desired behavior of the de-
signer: to have the robot react more, the closer it is. But it shares the prob-
lem of the constant magnitude proﬁle in the sharp transition to 0.0 velocity.
Therefore, another proﬁle might be used to capture the need for a strong
EXPONENTIAL DROP reaction but with more of a taper. One such proﬁle is a exponential drop off
OFF function, where the drop off is proportional to the square of the distance: for
every unit of distance away from the object, the force on the robot drops in
half. The exponential proﬁle is shown in Fig. 4.14c.
As can be seen from the previous examples, almost any magnitude proﬁle
is acceptable. The motivation for using magnitude proﬁles is to ﬁne-tune the
behavior. It is important to note that the robot only computes the vectors
acting on it at its current location. The ﬁgures display the entire ﬁeld for
all possible locations of the robot. The question then arises as to why do
the ﬁgures show an entire ﬁeld over space? First, it aids visualizing what
the robot will do overall, not just at one particular time step. Second, since
ﬁelds are continuous representations, it simpliﬁes conﬁrming that the ﬁeld is
correct and makes any abrupt transitions readily apparent.

4.4.3 Potential ﬁelds and perception

In the previous examples, the force of the potential ﬁeld at any given point
was a function of both the relative distance between the robot and an ob-
ject and the magnitude proﬁle. The strength of a potential ﬁeld can be a
function of the stimulus, regardless of distance. As an example recall from
Ch. 3 the feeding behavior of baby arctic terns where the feeding behavior
is guided by the stimulus “red.” This can be modeled by an attractive ﬁeld.
The bigger and redder an object in the baby’s ﬁeld of view, the stronger the
attraction, suggesting that a magnitude proﬁle using an increasing exponen-
tial function. Another important point that has already been mentioned is
that potential ﬁelds are ego-centric because robot perception is ego-centric.

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